Basic Ship Propulsion Ghose and Gokarn

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BASIC SHIP PROPULSION

J.P. Ghose

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R.P. Gokarn

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Formerly with Department of Ocean Engineering and Naval Architecture

Indian Institute 'of Technology

Kharagpur

.1£1

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Dedicated

to OUT Teachers:

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. Professor S.C. Mitra (b. 1914) Professor S.D. Niganl (b. 1924) Professor T.S. Raghuram (b. 1928)

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Acknowledgements

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The authors acknowledge their debt to the students of the Department of Naval Architecture and Ocean Engineering, Indian Institute of Technology, Kharagpur, who provided the motivation for writing this book. The authors would also like to acknowledge the support and encouragement they received from their colleagues in the Department. Professor O.P. Shagave valuable guidance in matters relating to the use of computers. Mi. R.K. Banik typed the mariuscript. The authors are deeply gra;teful to them.

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The manuscript of this book was initiallywritten by Professor Ghose, who wishes to acknowledge the financial support received from the University Grants Commission. Professor Chengi Kuo of Strathc1yde University re viewed Professor Ghose's manuscript and made some suggestions, for which the authors are very gratefuL The book was then completely revised and rewritten by Professor Gokarn taking into account Professor Kuo's com ments as well as the comments of an expert who reviewed Professor Ghose's manuscript. The authors are greatly indebted to Allied Publishers Lim ited and particularly to Mr. Suresh Gopal, Publishing Consultant, for their patience, support, encouragement and guidance during the period that the book was being rewritten.

Copyright Acknowledgements A book such as this leans heavily on the work of others, and the authors gratefully acknowledge their debt to the writers of the publications listed in the Bibliography. Specific thanks are due to the following for permission to reproduce copy right material: vii

Basic Ship Propulsion

viii

1. Dover Publications Inc., New York, for Tables A2.1, A5.1, A5.2 and

A5.3.

, )

2. The Indian Register of Shipping, Mumbai, for Equation 7.41 and Table 7.6. 3. *The International Organisation for Standardisation (ISO), Geneva, Switzerland, for Table 11.3. 4. International Shipbuilding Progress, Delft, The Netherlands, for Equa tions 9.1, 9.37, A3.1, A3.2, A3.3, A3.4, A3.7, A3.8, A3.9, A3.1O, A4.13, A4.14, A4.25, A4.26, A4.27, A4.33, A4.34 and A4.35, Figures 4.5, 4.6 and 4.7, and Tables 4.2,9.6, A3.1, A3.2, A3.3, A3.4, A3.5, A3.6, A3.8, A3.9, A3.10, A3.11, A3.12 and A4.3. 5. *The International Towing Tank Conference for Equations 8~8, 8. 9, 8.10, 8.11, 8.25. 8.30 and 8.31. 1

6. *Lloyd's Register of Shipping, Londpn, for Equation 7.39, Figur~ 9.1 and Tables 7.2, 7.3 and 7.4. I 7. The Royal Institution of Naval Architects, London, for Equations A4.12, A4.24 and A4.32, Figure 4.4 and Table 4.1. 8. *The Society of Naval Architects and Marine Engineers, New Jersey, for Equations 9.31, 9.38, 11.13, 11.20, 11.21, 11.22, 11.23, A4.3, A4.4, A4.5, A4.6, A4.17, A4.18, A4.19, A4.28 and A4.29, Figures 9.2 and 12.11, and Table 9.5. *Some of the organisations which have given permissi~n to reproduce copy right material require the following to be explicitly stated: 1. International Organisation for Standardisation:

Table 11.3 - Summary of manufacturing tolerances for ship propellers taken from ISO 484/2:1981 has been reproduced with the permission of the International Organisation for Standardisation, ISO. This standard can be obtained from any member body or directly from the Central Secretariat, ISO, Case postal 56, 1211 Geneva 20, Switzerland. Copy right remains \\lith ISO.

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Acknowledgements

ix

2. International Towing Tank Conference: The ITTC cannot take any responsibility that the authors have quoted the latest version andjor quoted correctly. 3. Lloyd's Register of Shipping: Equation 7.39, Figure 9.1 and Tables 7.2, 7.3 and 7.4 of this publication reproduce matter contained in the qoyd's Register of Shipping Rules and Regulations for the Classification of'ShipsjRules for the Mainte nance, Testing and Certification of Materials produced under licence from Lloyd's Register of Shipping, 71 Fenchurch Street, London, Eng land EC3M 4BS. 4. Society of Naval Architects and Marine Engineers: Equations 9.31, 9.38, 11.13, 11.20, 11.21, 11.22, 11.23, A4.3, A4.4, A4.5, A4.6, A4.17, A4.18, A4.19, A4.28, and A4.29, Figures 9.2 and 12.11, Table 9.5 and parts of Section 12.14 are reproduced with the permission of the Society of Naval Architects and Marine Engineers (SNAME). Material originally appearing in SNAM~ publications can not be reproduced without the written permission from the Society, 601 Pavonia Avenue, Jersey City, NJ 7306, USA. The authors are grateful to the following individuals for their help in get ting copyright permissions: 1. Ms. Pam Cote and Mr. John Grafton, Dover Publications Inc. 2. Mr. D.G. Sarangdhar, Chief Surveyor, Indian Register of Shipping. 3. Mr. Jacques-Olivier Chabot, Director (General Services and Market ing, International Organisation for Standardisation).

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4. Ir. J.H. Vink, Chief Editor, International Shipbuilding Progress. 5. Admiral U. Grazioli, Chairman, and Dr. E. De Bernardis, Secretary, 23 rd Executive Committee, International Towing Tank Conference.

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6. Mr. K. Neelakantan, Administrative Manager for India and Sri Lanka, Lloyd's Register of Shipping. 7. Mr. Trevor Blakeley, Chief Executive, The Royal Institution of Naval Architects. 8. Ms. Susan Grove Evans, Publications Manager, The Society of Naval Architects and Marine Engineers. The authors would further like to add that the equations, figures and tables taken from previously published books and papers have been modified, where necessary to conform to the format of this book. Any errors resulting from these modifications are the sole responsibility oBhe authors.

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Preface

In our long experience of teaching the subject of Ship Propulsion to un dergraduate and postgraduate students of Naval Architecture at the Indian Institute of Technology, Kharagpur, we have often felt the need for a basic text which would describe adequately the essential elements of ship propul sion. This book attempts to fulfil this need. "Basic Ship Propulsion", as implied in its title, deals with the fundamentals of ship propulsion. How ever. an attempt has also been made to cover the subject comprehensivelY:' A bibliography is provided for those readers who wish to pursue particular topics in greater detail and to an advanced level. A special feature of this book is the large number of examples and problems. These examples and problems have been specially designed to illustrate the principles described in the text and to aid the reader in understanding the subject. Chapter 1 introduces the subject of ship propulsion beginning with a short description of ships and ship propulsion machinery. The various propulsion devices used in ships are briefly reviewed. Chapter ~ considers the termi J nology and geometry of screw propellers, which are the dominant form of propulsion device used in ships today. The theory of propellers is discussed next in Chapter 3. Chapter 4 describes the behaviour of a propeller in undisturbed ("open") water and the methods ofrepresenting propeller open water characteristics, including those of methodical propeller series. The behaviour of a propeller when fitted in its customary position at the stern of a ship, and the resulting hull propeller interaction are discussed next in Chapter 5. Chapter 6 deals with the phenomenon of propeller cavitation. The topic of propeller blade strength is considered in Chapter 7. Propulsion experiments using models are described in Chapter 8. Chapter 9 deals with the important topic of propeller design, in which methods using experimen xi

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xii

tal model data and methods based on propeller theory are both considered. Speed trials and service performance of ships are discussed in Chapter 10. Chapter 11 deals with some miscellaneous topics concerning screw propellers. The last chapter of the book describes ship propulsion devices other than conventional propellers Each chapter, except the first, includes examples and problems based on the material covered in that chapter. The SI system of units has been used throughout the book, although for historical reasons there are occasional references to the British system. A notable exception to the use of SI units in this book is the unit of speed which, in conformity with accepted marine 1852 m per hour, 1 British knot:: practice, is the knot (1 metric knot 6080 ft per hour = 1853.1 m per hour; the metric knot has been used as the unit of speed along 'with m per sec). '

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In applying the principles of ship propulsion discussed in this book, it is now usual to make extensive use of computers. Although we make full use of computers in our work, we feel that the fundamentals are best learnt without undue reliance on computers. Therefore, there are only occa,sional references to computers in this book. Almost all the problems may be'solved without using computers. However, the reader may save consider::ble time and effort by using a "spreadsheet" for those problems involving tabular calculations. We have solved many of the problems and examples using Microsoft "Excel". We have also preferred t~ give much of the data required for designing propellers and similar tasks in the form of equations or tables rather, than as design charts. This should facilitate the use of computers for these tasks. Some useful data are given in the appendices at the end of the book, and there is a glossary of technical terms to help a reader unfamiliar with terms commonly used by naval architects and marine engineers.

1

Glossary Added inertia

The difference between the "virtual inertia" and the actual inertia of a body undergoing angular acceleration in a fluid. The virtual inertia is the ratio of a moment ,applied to. the body and the resulting angular acceleration.

Added mass

The difference between the "virtual mass" and the actual mass of a body undergoing linear ac celeration in a fluid. The virtual mass is the ratio of a force applied to the body and the resulting acceleration.

Amidships

The mid-length of the ship.

Anchor windlass

A device to raise the anchor of a ship.

Appendages

Small attachments to the hull of a ship, e.g. bilge keels and rudders.

Auxiliaries

Equipment necessary to allow the main equip ment, e.g. the main engine in a ship, to function effectively.

Ballast

A load placed in a ship to bring it to a desired condition of draught, trim and stability.

Bilge keels

Small projections fitted to the bottom corners (bilges) of a ship to reduce its rolling (oscillation about a longitudinal axis). xiii


xiv Block coefficient

The ratio of the volume of water displaced by the ship (displacement volume) and the volume of a rectangular block having the same length and breadth as the ship and a height equal to the draught of the ship.

Bollard

A fitting on a ship, pier or quayside to which mooring ropes may be attached.

Bossings

Longitudinal streamlined attachments to the hull

of a ship to support the propeller shafts in a twin

screw ship.

Boundary layer

A thin layer adjacent to the surface of a body

moving in a viscous fluid to which the viscous

effects are almost entirely confined.

Bow

The forward (front) part of the ship.

Crash stop

A manoeuvre in which a ship moving at full speed

is stopped and its direction of motion -reversed

as quickly as possible, normally by stopping the

. propeller revolution in one direction and starting

it in the opposite direction.

Damping

The 'phenomenon by which the amplitude of an , oscillation decreaSes with time and dies out.

Displacement volume

The volume of water displaced by a floating ship.

Displacement

The mass of water displaced by a floating ship,

equal to the mass of the ship.

Draught

The vertical distance between the bottom (keel)

of the ship and the surface of water in which the

ship is floating.

Hovercraft

~.

.

The popular name for an air cushion vehicle in

which a cushion of air beneath the vehicle sup

ports its weight.

I I

Glossary

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xv

Hull

The main body of a ship to which are attached superstructures and appendages.

Hydrofoil craft

A high speed marine craft in which the weight of the craft at high speed is entirely supported by the lift of hydrofoils (attachments like aircraft wings) fitted to the craft below the hull.

Laminar flow

A flow in which the,fluid appears to move in a series of thin sheets (laminae).

Longitudinal centre of buoyancy

The longitudinal coordinate of the centre of buoy ancy (centroid of the underwater volume) of the ship, often measured from amidships as a per- . centage of the length of the ship.

Midship coefficient

The ratio of the area of the immersed midship section of the ship to the product of its breadth and draught.'

Pitching

An angular oscillation of the ship about a trans verse axis.

Prismatic coefficient

The ratio of the displacement volume of the ship and the volume of a "prism" having a cross sec tional area equal to the area of the maximum immersed cross section (maximum section) of the ship and a length equal to the length of the ship.

Rolling

An angular oscillation of the ship about a longi tudinal axis.

Rudder

A device for ste.ering and manoeuvring a ship, consisting usually of a wing-like shape in a ver tical plane capable of being turned from side to side about a vertical axis.

Stern

The after (rear) part of the ship.

Torsionmeter

A device to measure the torque being transmitted by the propeller shafting in a ship.

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! xvi


Towboat

A ship designed to tow other ships.

Trawler

A fishing vessel which drags fishing gear.

Trim

The difference between the draughts forward and aft of a ship.

'lUg

A small ship meant for pushing or pulling a large ship that is not capable of moving safely on its own.

'lUrbulent flow

A flow in which, in addition to the average motion -of the fluid, there are small random movements of the fluid particles in all directions.

Waterline

The line of intersection of the surface of water and the hull of the ship,

Waterplane

The intersection of the surface of water in which a ship is floating and the hull of the ship.

Waterplane coefficient

The ratio of the area of the waterplane' to the product of the length and breadth of the ship.

Wetted surface

The area .of the outer surface of the ship hull in contact with the water.

L$

Nomenclature

Coordinates Cylindrical polar coordinates (r, 0, z) have been used for defining propeller geometry, with r along the radius, 0 being measured from the upwardly directed vertical and the z-axis coinciding with the propeller axis. The reference axis of a propeller blade is taken to coincide with the = 0 line, i.e. the blade is pointing vertically up. The x-axis is thus vertical and positive upward, the y-axis horizontal and positive to the right (for a right hand propeller) and the z-axis positive forward, the axes forming a right hand system. The origin of coordinates is at the intersection of the propeller axis and the blade reference line.

e

For a blade section, the origin is taken at the leading edge, the x-axis being . along the chord positive towards the trailing edge and the y-axis positive from the face to the back of the section. The principal axes of the section are denoted as the Xo- and Yo-axes.

Subscripts The subscripts 111 and S refer to the model and the ship respectively. Other subscripts have been defined in the text. xvii

Basic Sbip Propulsion

xviii

Symbols a

Area of blade section Axial inflow factor

al

Tangential inflow factor

A

Area

AD

Developed blade area

AE

Expanded blade area

AJ

Area of jet cross section

Ao

AI0

Propeller disc area

Ap

Projected blade area

AT

'Transverse projected area of ship above water

BP

Bollard pull

Immersed disc area of surface propeller

Bp

Taylor delivered power coefficient \

C

Added mass coefficient Blade section chord

Cmax

Maximum chord (width) of a blade

CA

Correlation allowance

C AA

Air and wind resistance coefficient

CB

Block coefficient

CD

Drag coefficient

CDR

Drag coefficient of a rough propeller surface

Nomenclature

J

xix

CDS

Drag coefficient of a smooth propeller surface

CF

Frictional resistance coefficient

CL

Lift coefficient

CN

Correlation factor for propeller revolution rate

Cp

Pressure coefficient

Cpmin

Minimum pressure coefficient

Cp

Mean pressure coefficient

Cp

Correlation factor for delivered /power Power coefficient

CR

Residuary resistance coefficient

CT

Total resistance coefficient

CT L

Thrust loading coefficient

CT Li

Ideal thrust loading coefficient

Cv

Viscous resistance coefficient

d

Boss diameter

D

Drag Propeller diameter

DD

Duct drag

DI

Pump inlet diameter

DA[

Momentum drag

e

Clearance between duct and propeller Eccentricity ratio

E

Modulus of elasticity

_


xx Euler number

I

Blade section camber

Frequency

fJ

Fundamental frequency of flexural vibration

It

Fundamental frequency of torsional vibration

F

Force Tangential force on the propeller Tow force in self

propulsion test

Fc

Centrifugal force

FH

Horizontal component of the tangential force

Fi

Tangential force on the i th blade

Fn

Froude number

Fv

Vertical 'component of the tangential force

g

Acceleration due to gravity

G:

Non-dimensional circulation 1Iodulus of rigidity

h

Depth of immersion of propeller shaft axis Depth of water Height of jet above waterline 11a.ximum tip immersion of surface propeller

hi

11usker's roughness parameter

H

Head of pump

hoss

Mass polar moment of inertia of boss

Ip

Mass polar moment of inertia of propeller

f f

Nomenclature

xxi

1xe

Second moment of area about the xo-axis of a blade section

1ya

Second moment of area .about the Yo-axis of a blade section

J

Advance coefficient Wind direction coefficient Lifting surface correction factor for angle of attack Lifting surface correction factor for camber

kD

Drag correction factor Inertia coefficient Mass coefficient Average propeller surface rpughness Average roughness of ship surface Correction factor for blade thickness Torque coefficient Torque coefficient in the behind condition Thrust coefficient Thrust coefficient in the behind condition Duct thrust coefficient

K'Q

Modified torqfIe coefficient of surface propeller

K'T

Modified thrust coefficient of surface propeller Duct length

L

Length of the ship

Lift


xxii Length dimension Noise level based on pressure Noise level based on power

m

Mass of fluid per unit time Mass of a blade

111

Mass of propeller

M

Mass dimension Mass of boss Bending moment due to torque Bending moment due to rake Bending moment due to skew Bending moment due to thrust Bending moment about the xo-axis of a blade section Bending moment about the Yo-axis of a blade section

n

Propeller revolution rate (revolutions per unit time) Torsion in a blade section due to torque Specific speed of pump Torsion in a blade section due to thrust

p

Pressure

p

Mean pressure

PA

Atmospheric pressure

I

t

.f'l

Nomenclature

I I

J

Pc

Roughness peak count per nun

Pressure due to cavitation

PV

Vapour pressure

Po .

Pressure without cavitation

P

Pitch of the propeller

p

Mean pitch of propeller

PB

Brake power

PD

Delivered power

PDO

Delivered power in open water

Pe

Effective pitch

PE

Effective power

PEn

Effective power of naked hull

PI

Indicated power

PJ

Power of waterjet

Ps

Shaft power

PT

Thr1..l.ll1t power

FTow

Towrope power

Fro

Thrust power in open water

q

Stagnation pressure

Q

Propeller torque

Pump discharge

Qi

Ideal torque

Torque of the i th blade

.xxiii


xxiv

Qo

Torque in open water

Q~

Vane wheel torque at inner radius

Q

v

, Vane wheel torque at outer radius

r

Radius of a blade section

f

Radius of blade centroid

rb

Boss radius

r Vi

Inner radius of vane wheel

rOV

Outer radius of vane w;heel

R

Propeller radius

R a 2.5

Root mean square roughness height in microns over a 2.5mm length I

RAA

Air and wind resistance

Rn

Reynolds number

Total resistance

RT , S

Se

S

Distance

Slip ratio

Span of a wing

Effective slip ratio Stress

Wetted surface of the ship

Wetted surface of bilge keels

Se

Compressive stress due to thrust and torque

S'e

Additional compressive stress due to centrifugal force

Nomenclature

xxv

.,. ST

Tensile stress due to thrust and torque

8'T

Additional tensile stress due to centrifugal force

Sc;(WE)

Encounter spectrum of seaway

t

Blade section thickness Thrust deduction fraCtion

i

Time

f

Mean thickness of blade

to

Blade thickness. extrapolated to

tl

Blade thickness at tip

T

Draught of the ship Propeller thrust

T

Time dimension

TD

Thrust of duct

Tc

Gross thrust of waterjet propulsion unit

Ti

Ideal thrust Thrust of the i th blade

To

Thrust in open water

TP

Towrope pull

Tp

Propeller thrust in a ducted propeller

rt

Vane wheel thrust at inner radius

T,0

Vane wheel thrust at outer radius

U

Induced velocity

Ua

Axial induced velocity

v

propelle~

axis


xxvi

Axial induced velocity of vane wheel at inner radius Axial induced velocity of vane wheel at outer radius Ut

Tangential induced velocity Tangential induced velocity of vane wheel at inner radius

U~v

Tangential induced velocity of vane wheel at outer radius

v

Velocity of flow.

v

Cavity volume Velocity induced due to duct

ii

Mean wake velocity

v'(r)

Average wake velocity at radius r

v(r,£I)

Velocity at the point (r, £I)

V

Characteristic velocity Ship speed Axial component of velocity

VA \

Speed of advance

VA

Average speed of advance

Vo

Speed of current

VG

Speed of ship over ground

VJ

Waterjet exit velocity

VK

Ship speed in knots

Vo

Observed ship speed Tangential velocity of propeller blade relative to water Relative wind velocity

Nomenclature

xxvii

VR

Resultant velocity

Vi

Tangential component of velocity

ViF

Speed of ship through water

VO.7R

Resultant velocity at 0.7 R

w

\Vake fraction

w

Average wake fraction

w'(r)

Wake fraction at radius r

w(rJJ)

Wake fraction at the point (r,O)

wefJ

Effective wake fractiop. !

W nom

Nominal wake fraction

W max

Local maximum wake fraction

wQ

Wake fraction (torque identity)

WT

\Vake fraction (thrust identity)

W

Weather intensity factor

Urn

\Veber number

x

Non-dimensional radius (r/R)

Overload fraction

Distance from the leading edge of a blade section

Non-dimensional boss radius Non dimensional radius of the blade centroid

Yc

oJ---

Distance normal to the axis between the centroids of the blade and the root section

xxviii

Basic Ship Propulsion "

Camber distribution of blade section

.

,

Thickness distribution of blade section

Zc

Distance parallel to the axis between the centroids of the blade and the root section

Z

Number of blades in propeller

Zv

Number of blades in vane wheel

1

+

k

Form factor

1

+

x

Load factor

Angle of attack

Duct dihedral angle

O:i

Ideal angle of attack

O:t

Angle of attack correction for blade thickness

0:0

No-lift angle \

{3

Duct exit angle Hydrodynamic pitch angle excluding induced velocities

{3[

Hydrodynamic pitch angle including induced velocities

'Y

Angle related to lift-drag ratio

r

Circulation

8

Taylor advance coefficient

8P

Increase in average power in waves

\

xxix

Nomenclature 6R

Resistance augment

6T

Thrust deduction

6.

Displacement of the ship Change in drag coefficient due to roughness Roughness allowance Correlation allowance for frictional resistance coefficient Change in lift coefficient due to roughness /

Change in torque coefficient dt,le to roughness Change in thrust coefficient due to roughness Pressure difference Increase in effective power due to winrl Speed correction for effect of wind Correlation allowance for wake fraction Rake angle Effective rake angle Average wave amplitude Efficiency

L

TlB

Propeller efficiency in the behind condition

TlD

Propulsive efficiency

f"JH

Hull efficiency

f"Ji

Ideal efficiency

xxx


7JiJ

Ideal jet efficiency

7JI

Inlet efficiency

7JJ

Jet efficiency

7JN

Nozzle efficiency

7JolJerall

Overall propulsive efficiency

7Jo

Propeller open water efficiency

7Jp

Pump efficiency

7JPO

Pump ~fficieri.cy in openwater (uniform inflow)

7JR

Relative rotative efficiency

7Js

Shafting efficiency

6

Angular position of the blade Relative wind direction off the bow

6s

Skew angle

.f~ i

!

,r !

t

I,

Coefficient of kinematic capillarity

Goldstein factor

Advance ratio

Scale ratio (ship dimension: model dimension)

Advance ratio including induced velocities

Coefficient of dynamic viscosity

Viscous correction factor

Propeller coefficient in the set (fl., CT, cp)

,

1

Nomenclature

xxxi

v

Coefficient of kinematic viscosity

P

Mass density of water

Pa

Density of air

Pm

Density of propeller material

(J

Cavitation numbers

Propeller coefficient in the set (fl, (J, tp)

(JO.7R

Cavitation number at O.7R

T

Ratio of propeller thrust to total thrust in a ducted propeller

Tc

Burrill's thrust loading coefficient

l

i

I I

I

I

I

Pitch angle

Propeller coefficient in the set (fl, (J, tp)

tPE

Effective skew angle

W

Angular velocity

WE

Wave encounter frequency Volume of displacement

1__

Physical Constants The following standard values have been used in the examples and problems: Density of sea water

= 1025 kg per m3

Density of fresh water

= 1000 kg per m3

Kinematic viscosity of sea water

= 1.188

Kinematic viscosity of fr~sh water

= 1.139 x 10-6 m 2

Acceleration due to gravity

= 9.81 m per sec 2

Atmospheric pressure

= 101.325 kN per m2

Vapour pressure of water

= 1.704kN per m 2

1 knot

= 0.5144m per sec

1hp

= 0.7457kW

xxxiii

J

_

X

10- 6 m2 per sec per sec

COlltents

v

DEDICATION

vii

JlCKNOWLEDGEMENTS

xi

PREFACE

xiii

GLOSSARY

xvii

NOMENCLATURE PHYSICAL CONSTANTS

xxxiii

GENERAL INTRODUCTION

1

1.1

Ships

1

1.2

Propulsion Machinery

2

1.3

Propulsion Devices

3

SCREW PROPELLERS

6

2.1

Description

6

2.2

Propeller Geometry

10

2.3

Propeller Blade Sections

15

2.4

Alternative Definition of Propeller Geometry

17

2.5

Pitch

19

2.6

Non-dimensional Geometrical Parameters

21

2.7

Mass and Inertia

24

CHAPTER

CHAPTER

1

2

xxxv


xxxvi

, . 'U} ".r','l

t'

PROPELLER THEORY

28

3.1

Introduction

28

3.2

Axial Momentum Theory

29

3.3

Momentum Theory Including Rotation

34

3.4

Blade Element Theory

39

3.5

Circulation Theory

45

3.6

Further Development of the Circulation Theory

56

THE PROPELLER

60

CHAPTER 3

CHAPTER 4

:

t,

••

~

"OPEN" WATER

,60

4.1

Introduction

4.2

Laws of Similarity

61

4.3

Dimensional Analysis

64

4.4

Laws of Similarity in Practice

66

4.5

Open Water Characteristics

74

4.6

Methodical Propeller Series

78

4.7

Alternative Forms of Propeller Coefficients

82

THE PROPELLER "BEHIND" THE SHIP

94

5.1

Introduction

94

5.2

Wake

95

5·3

Thrust Deduction

99

5.4

Relative Rotative Efficiency

100

5.5

Power Transmission

103

5.6

Propulsive Efficiency and its Components,

lOp

5.7

Estimation of Propulsion Factors

111

PROPELLER CAVITATION

115

6.1

The Phenomenom of Cavitation

115

6.2

'Cavitation Number

119

6.3

Types of Propeller Cav.itation

121

CHAPTER 5

..

IN

CHAPTER 6

. !

I i

t

~

Contents

xxxvii

6.4

Effects of Cavitation

124

6.5

Prevention of Cavitation

126

6.6

Cavitation Criteria

128

6.7

Pressure Distribution on a Blade Section

135

CHAPTER 7

STRENGTH OF PROPELLERS

140

7.1

Introduction

140

7.2

Bending Moments due to Thrust and Torque

142

7.3

Bending Moments due to Centrifugal Force

148

7.4

Stresses in a Blade Section

151

7.5

Approximate Methods

155

7.6

Classification Society Requirements

163

7.7

Propeller Materials

166

7.8

Some Additional Considerations

167

CHAPTER 8

PROPULSION MODEL EXPERIMENTS

179

8.1

Introduction

179

8.2

Resistance Experiments

180

8.3

Open Water Experiments.

185

8.4

Self-propulsion Experiments

190

8.5

\\Take Measurements

201

8.6

Cavitation Experiments

207

PROPELLER DESIGN

216

CHAPTER 9

9.1

Propeller Design Approaches

216

9.2

General Considerations in Propeller Design

217

9.3

Propeller Design using Methodical Series Data

222

9.4

Design of Towing Duty Propellers

236

9.5

Propeller Design using Circulation Theory

250


xxxviii

CHAPTER

10 SHIP 'lluALs

AND

SERVICE

PERFORMANCE

10.1

Introduction

277

10.2

Dock Trials

278

10.3

Speed Trials

279

10.4

Bollard Pull Trials

294

10.5

Service Performance Analysis

294

CHAPTER

11 SOME MISCELLANEOUS ToPICS

3

"

277

.~

·;1

I

I

I

I

I

308 ,

I

j

,

11.1

Unsteady PropelleJ.: Loading

308

11.2

Vibration and Noise

319

11.3

Propulsion in a Seaway

327

I

\

11.4

Propeller Roughness

331

1

11.5

Propeller Manufacture

338

11.6

Acceleration and Deceleration

,343

11.7

Engine-Propeller Matching

CHAPTER

12 UNCOl'.'VENTIONAL PROPULSION DEVICES

!

·347 359

12.1

Introduction

359

12.2

Paddle Wheels

360

12.3

Controllable Pitch Propellers

365

1204

Ducted Propellers

369

12~5

Supercavitating Propellers

386

12.6

Surface Propellers

391

12.7

Contra-rotating Propellers

401

12.8

Tandem Propellers

402

12.9

Overlapping Propellers

404

12.10

Other Ivlultiplc Propeller Arrangements

406

12.11

Vane Wheel Propellers

408

12.12

Other Unconventional Screw Propellers

410

,,

I

\

I

I

I

I

I

I

;

! ,

I i

Contents

xxxix

12.13

Cycloidal Propellers

412

12.14

Waterjet Propulsion

420

12.15

Flow Improvement Devices

431

12.16

Design Approach

436

ApPENDICES

l

1

Some Properties of Air and Water

440

2

Aerofoil Sections used in Marine Propellers

442

3

Propeller Methodical Series Data

444

4

Propulsion Factors

465

5

Propeller Blade Section Pressure Distribution

477

6

Goldstein Factors

482

7

Cavitation Buckets

484

8

Lifting Surface Correction Factors

485

REVIEW QUESTIONS

489

MISCELLANEOUS PROBLEMS

496

ANSWERS TO PROBLEMS

514

BIBLIOGRAPHY

540

INDEX

553

>.i·

,.>,.:'

CHAPTER

1

General Introduction (

1.1

Ships

The Earth may be regarded as a "water planet" , since 71 percent of its sur face is covered by water having an average depth of 3.7 km. Transportation across the oceans must therefore have engaged the. attention of humankind since the dawn of history. Ships started thousands of years ago as si~ple logs or bundles of reeds and have deve19ped into the huge complicated ves sels of today. \"ooden sailing ships are known to have appeared by about 1500 BC and had developed into vessels sailing around the world by about 1500 AD. Mechanical propulsion began to be used in ships by the beginning of the 19th Century, and iron followed by steel gradually took the place of wood for building large oceangoing ships, with the first iron-hulled ship, the "Great Britain", being launched in 1840. Ships today can be characterised in several ways. From the point of view of propulsion, ships may be either self-propelled or non-propelled requiring external assistance to move from one point to another. Ships may be ocean going or operating in coastal waters or inland waterways. Merchant ships which engage in trade are of many different kinds such as tankers, bulk carri ers, dry cargo ships, container vessels and passenger ships. Warships may be divided into ships that operate on the surface of water such as frigates and aircraft carriers, and ships that are capable of operating under water, viz. submarines. There are also vessels that provide auxiliary services such as tugs and dredgers. Fishing vessels constitute another important ship type. 1

L


2

Most of these types of ships have very similar propulsion arrangements. Hm\'ever, there are some types of very high speed vessels such as hovercraft and hydrofoil craft that make use of unconventional propulsion systems.

1. 2

Propulsion Machinery

For centuries, ships were propelled either by human power (e.g. by oars) or by ''lind power (sails). The development of the steam engine in the 18th Century' led to attempts at using this new source of power for ship propulsion, and the first steam driven ship began operation in Scotland in 1801. The early steam engines were of the reciprocating type. Steam was produced in! a boiler from raw sea water using wood or coal as fuel. Gradual advances in steam propulsion plants took place during the 19th Century, including the use of fresh water instead of sea water and oil instead of coal, improvements in boilers, the use of condensers and the development of compound steam engines. Reciprocating steam engines were' widely used for ship propulsion till the early years of the 20th Century, but have since then been gradually superseded by steam turbines and diesel engines. ' The first marine steam turbine was fitted in the vessel "'I'urbinia"in 1894 by Sir Charles Parsons. Since then, steam turbines have completely replaced reciprocating steam engines in steam ships. Steam turbines produce less vibration than reciprocating engines, make more efficient use of the high steam inlet pressures and very low exhaust pressures available with modern steam ge,nerating and condensing equipment, and can be designed to produce very high powers. On the other hand, turbines run at very high speeds and cannot be directly connected to ship propellers; nor can turbines be reversed. This makes it necessary to adopt special arrangements for speed reduction and reversing, the usual arrangements being mechanical speed reduction gearing and a special astern turbine stage, or a turbo-electric drive. These arrangements add to the cost and complexity of the propulsion plant and also reduce its efficiency. Since its invention in 1892, the diesel engine has continued to grow in popularity for usc in ship propulsion and is today the most common type of engine used in ships. Diesel engines come in a wide range of powers ~nd speeds, arc capable of using low grade fuels, and are comparatively efficient.

I J

I

I i

t

,ff J

I

I

1

I \ I

I j

I j

! i

\ .~

I, t

~

I

\ t

!.

'

.

t

-----------~

General Introduction

3

Low speed diesel engines can be directly connected to ship propellers and can be reversed to allow the ship to move astern. Another type of engine used for ship propulsion is the gas turbine. Like the steam turbine, the gas turbine runs at a very high speed and cannot be reversed. Gas turbines are mostly used in high speed ships where their low weight and volume for a given power give them a great advantage over"other types of engines. Nuclear energy has been tried for ship propulsion. The heat generated by a nuclear reaction is used to produce steam to drive propulsion turbines. However, the dangers of nuclear radiation in case of an accident have pre vented nuclear ship propulsion from being used in non-combatant vessels exc~pt for a few experimental ships such as the American- ship "Savannah", the German freighter "Otto Hahn"and the Russian icebreaker "V.l. Lenin". Nuclear propulsion has been used in large submarines with great success because nuclear fuel contains a large amount of energy in a very small mass, and because no oxygen is required for gen~rating heat. This enables a nu clear submarine to travel long distances under water, unlike a conventional submarine which has to come to the surface frequently to replenish fuel and air for combustion.

In addition to the conventional types of ship propulsion plant discussed in the foregoing, attempts are being made to harness renewable and non polluting energy sources such as solar energy, wind energy and wave energy for ship propulsion and to develop advanced technologies such as supercon ductivity and magneto-hydrodynamics. However, these attempts are still in a preliminary experimental stage.

1.3

Propulsion Devices

Until the advent of the steam engine, ships were largely propelled by oars imparting momentum to the surrounding water or by sails capturing the energy of the wind. The first mechanical propulsion device to be widely used in ships was the paddle wheel, consisting of a wheel rotating about a transverse axis with radial plates or paddles to impart an astern momentum to the water around the ship giving it a forward thrust. The early steam ers of the 19 th Century were all propelled by paddle wheels. Paddle wheels

4


are quite efficient when compared with other propulsion devices but have several drawbacks including difficulties caused by the variable immersion of the paddle wheel in the different loading conditions of the ship, the increase in the overall breadth of the ship fitted with side paddle wheels, the inabil ity of the ship to maintain a steady course when rolling and the need for slow running heavy machinery for driving the paddle wheels. Paddle wheels were therefore gradually superseded by screw propellers for the propulsion of oceangoing ships during the latter half of the 19th Century. The Archimedean screw. had been used to pump water for centuries, -and proposals had been made to adapt it for ship propulsion by using it to impart momentum to the water at the stern of a ship. The first actual use of a screw to propel a ship appears to have b~en made in 1804 by the American, Colonel Stevens. In 1828, Josef Ressel of Trieste successfully used a screw propeller in an 18 m long experimental steamship. The first practical applications of screw propellers were made in 1836 by Ericsson in America and Petit Smith in England. Petit Smith's propeller consisted of a wooden screw of one thread and two complete turns. During trials, an accident caused a part of the propeller to break off and this surprisingly led to an incr~8.se in the speed of the ship. Petit Smith then improved the design of his propeller by decreasing the width of the blades and increasing the number of threads, producing a screw very similar to modern marine propellers. The screw propeller has since then become the predominant propulsion device used in shipl3. Certain variants of the screw propeller are used for special applications. One such variant is to enclose the propeller in a shroud or nozzle. This improves the performance of heavily loaded propellers, such as those used in tugs. A controllable pitch propeller allows the propeller loading to be varied over a wide range without changing the speed of revolution of the propeller. It is also possible to reverse the direction of propeller thrust without chang ing the direction of revolution. This allows one to use non-reversing engines such as gas turbines. When propeller diameters are restricted and the pro pellers are required to produce large thrusts, as is the case in certain very high speed vessels, the propellers are likely to experience a phenomenon called "cavitation", which is discussed in Chapter 6. In cjrcumstances where extensive cavitation is unavoidable, t.he propellers are specially designed to

L

General Introduction

5

operate in conditions of full cavitation. Such propellers are popularly known as "supercavitating propellers". Problems due to conditions of high propeller thrust and restricted diame ter, which might lead to harmful cavitation and reduced efficiency, may be avoided by dividing the load between two propellers on the same shaft. Mul tiple propellers mounted on a single shaft and turning in the same direction are called "tandem propellers". Some improvement in efficiency can be 0 b tained by having the two propellers rotate in opposite directions on coaxial shafts;I Such "contra-rotating propellers"are widely used in torpedoes. Two other ship propulsion devices may be IIl;entioned here. One is the vertical axis cycloidal propeller, which consists 6f a horizontal disc carrying a number of vertical blades projecting below W. As the disc rotates about a vertical· axis, each blade is constrained to t.urn about its own axis such that all the blades produce thrusts in the same direction. This direction can be controlled by a mechanism for setting the positions of the vertical blades. The vertical axis propeller can thus produce a thrust in any direction, ahead, astern or sideways, thereby greatly/improving the manoeuvrability of the vessel. The second propulsion device that may be mentioned is the waterjet. Historically, this is said to be the oldest mechanical ship propulsion device, an English patent for it having been granted to Toogood and Hayes in 1661. In waterjet propulsio~, as used today in high speed vessels, an impeller draws water from below the ship and discharges it astern in a high velocity jet just above the surface of water. A device is provided by which the direction of the waterjet can be controlled and even reversed to give good manoeunability. Waterjet propulsion gives good efficiencies i1\ high speed craft and is becoming increasingly popular for such craft. Because of their overwhelming importance in ship propulsion today, this book deals mainly with screw propellers. Other propulsion devices, including variants of the screw propeller, are discussed.in Chapter 12.

,

l__~

I

.f

\

CHAPTER

2

Screw Propellers 2.1

Description

A screw propeller consists of a number of blades attached to a hub or boss, as shown in Fig. 2.1. The boss is fitted to the propeller shaft throughwhich the power of .the propulsion machinery of the ship is transmitted to the propeller. When this power is delivered to the propeller, a turning ,moment or torque Q is applied making the propeller revolve about its axis with a speed ("reyolution rate") n, thereby producing an axial forc~ or thrust T causing the propeller to move forward with respect to the surrounding medium (water) at a speed of advance VA. The units of these quantities in the 81 system are:

I

I, \

Q" n

T VA

I

Newton-metres revolutions per second Newtons metres per second

I

I (

The revolution rate of the propeller is often given in terms of revolutions per minute (rpm), and the speed of advance in knots (1 knot = 0.5144 metres per second). The point on the propeller blade farthest from the axis of revolution is called the biade tip. The blade is attached to the propeller boss at the root. The surface of the blade that one would see when standing behind 6

7

Seren' Propellers

DIRECTION OF REVOLUTION

FOR AHEAD MOTION

TIP LEADING EDGE .'

TRAILING EDGE

PROPELLER AXIS

. L .-' ...... --.,..,.,..

.-' .-'

PROPELLER SHAFT

Figure 2.1 : A Three-Bladed Right Hand Propeller.

the ship and looking at the propeller fitted at the stern is called the face of the propeller blade. The opposite surface of the blade is called its back. A propeller that revolves in the clockwise direction (viewed from aft) when propelling the ship forward is called a right hand propeller. If the propeller turns anticlockwise when driving the ship ahead, the propeller is left handed. The edge of the propeller blade which leads the blade in its revolution when the ship is being driven forward is called the leading edge. The other edge is the trailing edge. (

'When a propeller revolves about its axis, its blade tips trace out a circle. The diameter of this circle is the propeller diameter D. . The number of p~opeller blades is denoted by Z. The face of the propeller blade either forms a part of a helicoidal or ss;rew ~urface, or is defined with respect to it; hence the name "screw propeller". A h'~iicoidal surface is generated when a line revolves about an axis while simultaneously advancing along it. A point on the line generates a three-dimensional curve called a helix. The distance that the line (or a point on it) advances along the axis in one revolution is called the pitch of the helicoidal surface (or the helix). The pitch of the


8

I !

RAKE ANGLE E

NO RAKE

RAKE AFT

(0) RAKE

i

.i

,

}

NO SKEW

MODERATELY

SKEINED

HEAVILY SKEWED

(b) SKEW

Figure 2.2 : Raile and Silew. \.

helicoidal surface which defines the face of a propeller blade is called the (face) pitch P of the propeller. If the line generating the helicoidal surface is perpendicular to the axis about which it rotates when advancing along it, the helicoidal surface and the propeller blade defined by it are said to have no rake. If, however, the generating line is inclined by an angle e to the normal, then the propeller has a rake angle e. The axial distance betwe~n points on the generating line at the blade tip and at the propeller axis is the rake. Propeller blades are sometimes raked aft at angles up to 15 degrees to increase the clearance (space) between the propeller blades and the hull of the ship, Fig. 2.2(a). '

., !

,

.,

Screw Propellers

9

Com-:ider the line obtained by joining the midpoints between the leading and trailing edges of a blade at different radii from the axis. If this line is straight and passes through the axis of the propeller, the propeller blades have no skew. Usually ho~ever, the line joining the midpoints curves towards the trailing edge, resulting in a propeller whose blades are skewed back. ~kew.. i,s~d to reduce vibration. Some modern propeller designs have heavily skewed blades. The angle Os between a straight line joining the centre of the propeller tq .iha..midRQinL~~JheWt and a line joining the centre and the midpoint at the blade tip is a measure of skew, Fig.2.2(b) . . \

Example 1 In a propeller of 4.0 m diameter and 3.0 m constant pitch, each blade face coincides with its defining helicoidal surface. The distance. of the blade'tip face from a plane normal to the axis is 263.3 mm, while the distance of a point on the face at the root section (radius 400 mm) from the same plane is .52.7 mm, both distances being measured in a plane through. the propeller axis: The midpoint of-the root section is 69.5'mm towards the leading edge from a plane through the propeller axis, while the blade tip is 1:285.6 mm towards the trailing ~dge from the same plane. Determine the rake and skew angles of thepropell~ The tangent of the rake angle is given by: tan£; =

difference in rake of the two sections difference in their radii

=

263.3 - 52.7 2000 - 400

= 0.131625 Rake angle

€

=

7.5 0

The angles which the midpoints of the root section and the tip make with the reference plane are given by: sineo

=

69.5 400

sin e1

=

-1285.6 = -0.64280 2000

=

0.17375

The skew angle is therefore (eo - ( 1 )

4

=

eo

50 0

=

10.00

0


10

2.2

Propeller Geometry

The shape of the blades of a propeller is usually defined by specifying the shapes of sections obtained by the intersection of a blade by coaxial right circular cylinders of different radii. These sections are called radial sections or cylindrical sections. Since all the Z propeller blades are identical, only one blade needs to be defined. It is convenient to use cylindrical polar coordinates (r, e, z) to define any point on the propeller, r being the radius measured from the propeller a.xis, an angle measured from a reference plane passiI).g through the axis, and z the distance from another reference plane normal to the axis. The z = 0 reference plane is usually taken to pass through the intersection of the propeller axis and the generating line of the helicoidal surface in the e = 0 plane. !

e

.,

,

Consider the section of a propeller blade by a coaxial circular cylinder of radius r, as shown in Fig.2.3(a). The blade is pointing vertically up. The figure also shows~the helix over one revolution defining the blade face at radius r, and the reference planes e = 0 and z = O. The projections; of this figure on a plane perpendicular to the propeller axis and on a hqrizontal plane are shown in Fig. 2.3(b) and (c). If the surface of the c:ylirider is now cut along the line AAl, joining the two ends of the helix, and the surface unwrapped into a plane, a rectangle of length 27rr and breadth P (the pitch of the helix) is obtained, the helix being transformed into the diagonal as shown in Fig. 2.3(d). The radial section takes the shape shown in the figure, and this shape is the expanded section at the radius r. The angle

Screw Propellers

11

of the leading and trailing edges on the base line of each section gives the expanded blade outline. The area within the expanded outlines of all the blades is the expanded blade area, AE.

HELIX A

(0)

(b)

(d)

(c)

CYLINDER. RADIUS r

z

t

r LA~¢ p

L--------I

1---2TIr----!

1 p

1

Figure 2.3: Propeller Blade Cylindrical Section.

Given the expanded blade outline and sections, it is quite simple to obtain the actual shape of the propeller blade as represented by its projections on three orthogonal planes. This is so because from the expanded outline and sections one readily obtains the cylindrical polar coordinates (r, (), z) of the leading and trailing edges, or indeed of any other point on the propeller blade

12


surface. It is usually convenient to transform these coordinates to Cartesian coordinates using the axes shown in Fig. 2.4: x = r cos (), y = r sin (), z = z. The blade outline projected on a plane normal to the z-axis is called the projected blade outline, and the area contained within the projected outlines of all the blades is called the projected blade area, Ap. x

IC

+

p

.

m-t (c) PROALE

(b) PROJECTED OUTLINE

DEVELOPED OUTUNE

(0) EXPANDED SECTIONS EXPANDED OUTUNE

DETAILS OF EXPANDED SECTION AT RADIUS r

T

. __ z

y

(d) PLAN

Figure 2.4 : Propeller Drawing.

Screw Propellers

13

The projections of the leading and trailing edges of a blade section on a plane tangential to the helix at the point C in Fig.2.3(d), Le. L' and T', are also associated '.'lith what is called the developed blade outline. If these tangent planes for the different radial sections are rotated through the pitch angles

°

L

14


blade departs from its base line at the different radii. The offset of the face above its base line is' variously called wash-back, wash-up, wash-away or setback, a negative offset (below the base line) being called wash-down. The offsets of the leading and trailing edges are called nose tilt and tail tilt. The distribution of pitch over the radius, if not constant, is shown separately. The variation of maximum blade thickness with radius r and the fillet radii where the blade joins the boss are also indicated. The internal details of the boss showing how it is fitted to the propeller shaft may also be given.

Example 2 In a propeller of 5.0 m diameter and 4.0 m pitch, radial lines from the leading and trailing edges of the section at 0.6R make angles of 42.2 and 28.1 degrees with the reference plane through the propeller iaxis. Determine the width of the expanded blade outline at 0.6R. The radius of the section at 0.6R,

r

=

0.6x

5

2=

The pitch angle at this section is given by: 'p 4 tanlp = = = 0.4244 21ir 21i X 1.5

1.5m

=

1500mm

cos lp = 0.9205

lp = 22.997° Referring to Fig. 2.3,

8T = 28.1 0 (given)

The width of the expanded outline at 0.6R is:

c

=

8L and 8T being in radians

so that, 1500 [42.2 c =

0

+ 28.1

57.3 0.9205

., "'.

0 ]

= 1999.2 nun

This assumes t.hat the section is flat faced, i.e. Land T in Fig. 2.3(c) coincide with L' and T' respectively.

Screw Propellers

15

Example 3 The cylindrical polar coordinates (r, B, z) of the trailing edge of a flat faced propeller blade radial section are (1500 nun, -30°, -400 nun). If the pitch of the propeller is 3.0 m, and the expanded blade width is 2000 mm, determine the coordinates of the leading edge. The leading and trailing edges of a radial section have the same radius, Le. r = 1500rnm. The pitch angle is given by: tan'P = -

P

21fr'

=

3000 21f x 1500

=

0.3183

cos'P = 0.9529

'P = 17.657°

sin'P = 0.3033

If the a coordinates of the leading and trailing edges are BL and BTl then the expanded blade width c is given by:

r((h - aT)'

.

c = --'-----'-,aee FIg. 2.3(c)

cos'P

Le.

2000 =

1500

[a£ -

.,

(-30 0 )!l57.3

--~=-0-.-:'95::-2-9'--':'::':""--

or Also,

ZL

=

ZT

+ c sin 'P

= -400 + 2000 x 0.3033 = 206.6 nun Le. the coordinates of the leading edge are (1500 rom, 42.80°, 206.6 mm).

2.3

Propeller Blade Sections

The expanded blade sections used in propeller blades may generally be di vided into two types: segmental sections and aerofoil sections. Segmental sections are characterised by a flat face and a circular or parabolic back, the maximum thickness being at the midpoint between the leading and trailing edges, the edges being quite sharp, Fig.2.5(a). In aerofoil sections, the face

16


-~_ ~ _ ---l~:===_..L1~-".L- _o::::-"-::.--_

(0) SEGMENTAL

(c) LENTICULAR

(b) AEROFOIL

SECTION

SECTIONS

SECTION"

Figure 2.5: Propeller Blade Sections.

'mayor may not be flat, the maximum thickness is usually nearer the lead ing edge, which is often more rounded than the trailing edge, Fig.2.5(b). More rarely, a propeller may have lens-shaped or lenticular blade sections, Fig.2.5(c)j such sections are used in propellers that are required to work equally efficiently for both direc'tions of revolution. CAMBER LINE

Yt

NOSE-TAIL UNE: X

i

x"

~ \

C

C

= chord

t "7 maximum thickness

y (x) t

=

f

= comber

thickness distribution

Figure 2.6: Definition of an Aerofoil Section.

An aerofoil section is usually defined in terms of the mean line or centre line between its lower and upper surfaces, Le. the face and the back, and a thickness distribution along its length, as shown in Fig. 2.6. The length of the section, or its chord c, is measured between the leading edge or nose and the trailing edge or tail,and the centre line is defined by its offsets Yc(x) from the nose-tail line at different distances x from the leading edge. The offsets of the face and back, Yt(x), are measured from the mean line perpendicular to it. The maximum offset of the mean line is the section camber f and the maximum thickness of the section is its thickness t. Mean lines and thickness

Screw Propellers

17

distributions of some aerofoil sections used in marine propellers are given in Appendix 2. Instead of measuring the section chord on the nose-tail line, it is usual in an expanded propeller blade section to define the chord as the projection of the nose-tail line on the base line, which corresponds to the helix at the given radius, i.e. the chord c is taken as L' T' rather than LT in Fig. 2.3(c) or Fig.2.4(a). If the resultant velocity of flow to a blade section is VR as shown in Fig.2.7, the angle between the base chord and the resultant velocity is I

NO-L1FjT LINE

+ LIFT

DRAG

t o

I -.10

DRAG

Figure 2.7 : Angle ofAttack.

called the angle of attack, a. The blade section then produces a force whose components normal and parallel to VR are the lift and the drag respectively. For a given section shape, the lift and drag are functions of the angle of attack, and for a certain (negative) angle of attack the lift of the section is zero. This angle of attack is known as the 'no-lift angle, Qo.

2.4

Alternative Definition of Propeller Geometry

When a propeller is designed in detail beginning with the design of the expanded sections, the geometry of the propeller is sometimes defined in a slightly different way. The relative positions of the expanded sections are indicated in terms of a blade reference line, which is a curved line in space that passes through the midpoints of the nose-tail lines (chords) of the sections at the different radii. A cylindrical polar coordinate system (r,e,z) is chosen as shown in Fig. 2.8. The z = 0 reference plane is normal to the propeller axis, the e = 0 plane passes through the propeller axis (z-axis),

-:1...

_

18

Basic Ship Propulsion 8=0 PLANE

z=o PLANE

PITCH OF HELIX = P

z A

SURFACE OF CYLINDER UNWRAPPED INTO A PLANE

z=o p

8=0

A

t'--------- ----------.t 2nr

Figure 2.8 : Alternative Definition of Propeller Geometry.

19

Screw Propellers

and both pass through the blade reference line at the boss. The pitch helix at any radius r passes through the 0 = 0 plane at that radius. The angle between the plane passing through the z-axis and containing the point on blade reference line at any radius and the 0 = 0 plane is the skew angle Os at that radius. The distance of the blade reference line at any radius from the z = 0 reference line is the total rake iT at that radius, and consists of the generator line rake ic and skew induced rake is as shown in Fig. 2.8. Rake aft and skew back (i.e. towards the trailing edge) are regarded as positive. This requires the pesitive z-axis to be directed aft. Rake and skew may be combined in such a way as to produce a blade reference line that lies in a single plane normal to the propeller axis. Warp is that particular combination of rake and ske~ that pro9.uces a zero value for the total axial displacement of the reference point of a propeller blade section. '

2.5

Pitch

As mentioned earlier, the face of a propeller blade is defined with respect to a helicoidal surface, the pitch of this surface being the face pitch P of the propeller. The helicoidal surface is composed of helices of different radii r from the root to the tip of the propeller blade. If all the helices have the same pitch, the propeller is said to have a constant pitch. If, however, the pitch of the helicoidal surface varies with the radius the propeller has a radially varying or variable pitch. (In theory, it is also possible to have circumferentially varying pitch when the ratio of the velocity of advance to the tangential velocay of the generating line of the helicoidal surface is not constant.) If P(r) is the pitch at the radius r, the mean pitch P of the propeller is usually determined by taking the "moment mean":

i

R

rb

P(r) 1'dr

(2.1)

being the radius at the root section where the blade joins the boss and R the propeller radius.

rb

1


20 ,

Consider a propel~er of diameter D and pitch P operating at a revolution rate n and advancing at a speed VA. If the propeller were operating in an unyielding medium, like a screw in a nut, it would be forced to move an axial distance nP in unit time. Because the propeller operates in water, the advance per unit time is only VA, i.e. the propeller slips in the water, the slip being nP - lAo The slip ratio is defined as: s

=

nP-VA nP

(2.2)

If VA = 0, B = 1 and the propeller operates in the 100 percent slip condition. If VA = nP, s = 0, and the propeller operates at zero slip. If the value of P used in Eqn. (2.2) is the face (nominal) pitch, s is the nominal slip ratio. However, at zero slip the thrust T of a propeller should be zero, and the effective pitch Pe may be determined in this way, i.e. by putting Pe = VAin for T = O. If the effective pitch is used in Eqn. (2.2), one obtains the effective slip ratio, Be. If in defining slip the speed of the ship V is used instead of the speed of advancel-A, one obtains the apparent slip. (V and l'A are usually not the same). Example 4 A propeller running at a revolution rate of 120 rpm is found to produce no thrust when its velocity of advance is 11.7knots and to work most efficiently when its velocity of advance is 10.0 knots. What is the effective pitch of the propeller and the effective slip ratio at which the propeller is most efficient? Ii' . < ectlve s p ratlo Eff

Se

\Vhen the propeller produces zero thrust,

e - VA = nP <, P n e

Se

= 0, and:

11. 7 x 0.5144 = 3.0092 m 120 60

\Vhen the propeller works most efficiently: Se

=

1- VA

nPe

10.0 x 0.5144' = 1 - 120 = 0.1453 60 x 3.0092

Screw PropeIlers

2.6

21

Non-dinlensional Geometrical Parameters

As will be seen in subsequent chapters, the study of propellers is greatly dependent upon the use of scale models. It is therefore convenient to define the geometrical and hydrodynamic characteristics of a propeller by non dimensional par.ameters that are independent of the size or scale of the pro peller. The major non-dimensional geometrical parameters used to describe a propeller are: - .Pitch ratio P/ D : the ratio of the pitch to the diameter of the propeller. - E:lI..-panded blade area ratio AE/Ao: the ratio of the expanded area of all the blades to the disc area Ao of the propeller, A o = 7l" D2 /4" (The developed blade area ratio AD/Ao and the projected blade are~' ratio Ap/Ao are similarly defined.) - Blade thickness fraction to/ D: the ratio of the maximum blade thickness extrapolated to zero radius, to, divided by the propeller diameter; see Fig.2.4(c). - Boss diameter ratio diD: the ratio of the boss diameter d to the propeller diameter; the boss diameter is measured as indicated in Fig.2.4(c) . . Aerofoil sections are also described in terms of non-dimensional parame ters: the camber ratio f Ic and the thickness-chord ratio tic, where c, f and t are defined in Fig. 2.6. The centre line camber distribution and the thickness distribution are also given in a non-dimensional form: Yc(x)lt and Yt(x)lt as functions of xl c.

Example 5 In a four-bladed propeller of 5.0m diameter, the expanded blade widths at the different radii are as follows:


22 r/R cnun

0.2 1454

0.3 1647

0.4 1794

0.5 1883

0.6 1914

0.7 1876

0.8 1724

0.9 1384

1.0

o

The thickness of the blade at the tip is 15 mm and at r/ R = 0.25, it is 191.25 rom. The propeller boss is shaped like the frustum of a cone with a length of 900 nun, and has forward and aft diameters of 890 nun and 800 mm. The propeller has a . rake of 15 degrees aft and the reference line intersects the axis at the mid-length of the boss. Determine the expanded blade area ratio, the blade thickness fraction and the boss diameter ratio of the propeller.

By drawing the profile (elevation) of the boss, and a line at 15 degrees fro~ its mid-length, the boss diameter is obta,ined as d == 834nun, giving a boss diameter ratio:

d D

==

834 == 0.1668 5000

By drawing the expanded outline (i.e. c as a function of r), the blade width at the root section (r/R == 0.1668 or r == 417mm) is obtained as 1390mm. The area within the blade outline from the root section to r/ R == 0.2 or r == 500 I1Un is' thus:

\

A 1 == 1390; 1454 (500 _ 417) == 118026 mm 2

The area of the rest of the blade may be obtained by Simpson's First Rule, according to which:

J

1

f(x)dx == - x s 3

n

X

L8Mi x f(xd . 1=1

where 1/3 is the common multiplier, s is the spacing between the n equidistant values of Xi, and 8M; are the Simpson Multipliers 1,4,2,4, ... ,4,2,4, Ii n must be an odd int(~ger. Here, c is to be integrated over the radius from 0.2R to LOR, the spacing between the radii being O.lR == 250 rom. The integration is usually carried out in a table as shown in the following:

23

Screw Propellers r

R

cmm

8M

j(A)

0.2

1454

1

1454

0.3

1647

4

6588

A 2 = ~ x 250 x 39478

0.4

1794

2

3588

= 3289833 mm2

0.5

1883

4

7532

0.6

1914

2

3828

0.7

1876

4

7504

0.8

1724

2

3448

0.9

1384

4

5536

1.0

0

1

-

0

39478

The expanded blade area of all the four blades is thus: I

AE

= 4 (118026 + 3289833)'mm2 = 13.6314 m 2

The disc area of the propeller is: Ao

= ~D2 = ~ X 5.0002 = 19.6350m2

The expanded blade area is therefore:

AE Ao

= 13.6314 = 0.6942 19.6350

The blade thickness extrapolated linearly to the shaft axis is:

_ _ tl - to.25 _ 5 _ 15 - 191.25 _ 250 mm to - tl 1 _ 0.25 - 1 1 _ 0.25 \ where to, tl and to.25 are the blade thicknesses at r / R The blade thickness fraction is therefore:

to D

I.

=

250 5000

= 0.050.

= 0, 1.0 and 0.25 respectively.

,,~

I

'f;~ ,a". Ba.<;ic Ship Propulsion

24

'I , I 1:',', ,':~

2.7

r ~i

Mass and Inertia

The mass of a propeller needs to be calculated to estimate its cost, and both the mass and the polar moment of inertia are required for determining the vibration characteristics of the propeller shafting system. The mass and polar moment of inertia of the propeller blades can be easily determined by integrating the areas of the blade sections over the radius. The mass and inertia of the boss must be added. Thus, one may write:

,i

R

M=Pm Z

adr+Mboss

(2.3)

rb

lp

= Pm Z

l'

R

2

ar dr

+ lboss

(2.4)

rb

where M and lp are the mass and polar moment of inertia of the propeller, Pm is the density of the propeller material, a the area of the blade section at radius r, and Mboss and lboss the mass and polar moment of inertia of the propeller boss, the other symbols having been defined earlier. . The area of a blade section depends upon its chord c and thickness t so that for a blade section of a given type, one may write: \

a = constant

X

ext

(2.5)

The ~hords or blade widths at the different propeller radii are proportional to the expanded blade area ratio per blade, while the section thicknesses depend upon the blade thickness fraction. One may therefore write: -"\1 -

AE to 3 kmPm Ao D D + Mboss

AE to 5 I p = ki Pm Ao D D + lhoss

(2.6) (2.7)

where k m and ki are constants which depend upon the shape of the propeller blade sections. '

25

Screw Propellers

Problems 1. A propeller of 6.0 m diameter and constant pitch ratio 0.8 has a flat faced

expanded section of chord length 489 nun at a radius of 1200 mm. Calculate the arc lengths at this radius of the projected and developed outlines. 2. The distances of points on the face of a propeller blade from a plane normal to the axis measured at the trailing edge and at 10degree intervals up to the leading edge at a radius of 1.75 m are found to be as follows:

Angle, deg Distance, mm:

TE -37.5 750

-30 770

-20 828

-10 939

o 1050

20 1272

10 1161

30 1430

LE 32.5 1550

The propeller has a diameter of 5.0 m. The blade section has a flat face except nea~ the trailing edge (TE) and the leading edge (LE). Determine the pitch at this radius. If the propeller has a constant pitch, what is its pitch ratio? 3. The cylindrical polar coordinates (r,B,z)of a propeller, r being measured in rom from the propeller axis, (j in degrees from a reference plane through the axis and z in rom from a plane normal Ito the axis, are found to be (1500, 10, 120) at the leading edge and (1500, -15, -180) at the trailing edge at the blade section at 0.6R. The blade ~ection at this radius has a flat face. Determine the width of the expanded outline at thi~ radius and the position of the reference line, B == 0, with respect to the leading edge. What is the pitch ratio of the propeller at 0.6R? The propeller has no rake. 4. The expanded blade widths of a three-bladed propeller of diameter 4.0 m and pitch ratio 0.9 are as follows:

rjR : cmm:

0.2

0.3

1477 1658

0.4 1808

0.5 1917

0.6 1976

0.7 1959

0.8 1834

0.9

1497

1.0

o

Find the expanded, developed and projected blade area ratios of the propeller•. Assume that the root section is at 0.2R, the blade outline is symmetrical and the blade sections are flat faced. 5. The face and back offsets of a propeller blade section with respect to a straight line joining the leading and trailing edges ("nose-tailline") are as follows:

Distance from leading edge

mm 0 50

L

Face offset

Back offset

nun 0 -24.2

rom 0

37.8

Basic Ship Propulsion Distance from leading edge mm 100.

200 300 400 500 600 700 800

900 1000

Face offset

Back offset

mill -32.4 -42.5 -48.0 -:..50.2 -49.4 -45.3 -38.3 -29.1 -19.2 -5.0

54.8 77.5 91.1 98;3 99.4 94.3 82.8 64.2 37.1 5.0

rom

Determine the thit::kness-chord ratio and the camber ratio of the section. 6. A propeller of a single screw ship has a diameter of 6.0 m and a radially varying pitch as follows:

r/R : 0.2

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 . P / D: 0.872 0.902 0.928 0.950 0.968 0.982 0.992 0.998 1.000

Calculate the mean pitch ratio of the propeller. What is the pitch at 0.7R? 7. ,

A-

propeller of 5.0 m diameter and 1.1 effective pitch ratio has a speed of advance of 7.2 m per sec when running at 120rpm. Determine its slip ratio. If the propeller rpm remains unchanged, what should be the speed of advance fot., the propeller to have (a) zero slip and (b) 100 percent slip?

8. In 'a four-bladed propeller of 5.0 m diameter, each blade has an expanded area of 2.16m2 • The thickness of the blade at the tip is 15mill, while at a radius of 625 inm the thickness is 75 mm with a .linear variation from root to tip. The boss diameter is 835 mill. The propeller has a pitch of 4.5 m. Determine the pitch ratio, the blade area ratio, the blade thickness fraction and the boss diameter ratio of the propeller. 9.

A crudely made propeller consists of a cylindrical boss of 200 mm diameter to which are welded three flat plates set at an angle of 45 degrees to a plane normal to the propeller axis. Each flat plate is 280 mm wide with its inner edge shaped to fit the cylindrical boss and the outer edge cut square so that the distance of its midpoint from the boss is 700 mID. Determine the diameter, the mean pitch ratio and the expanded and projected blade area ratios of this propeller.

Screw Propellers

27

10. A three-bladed propeller of diameter 4.0m has blades whose expanded blade widths and thicknesses at the different radii are as follows:

r/R Width, mm Thickness, mm ;

0.2 1000 163.0

0.3 1400 144.5

0.4 1700 126.0

0.5 1920 107.5

0.6 2000 89.0

0.7 1980 70.5

0.8 1800 52.0

0.9 1.0 1320 0 33.5' 15.0

The blade sections are all segmental with parabolic backs, and the"boss may be regarded as a cylinder of length 900mm and inner and outer diameters of 400 mm and 650 mm respectively. The propeller is made of Aluminium Nickel Bronze of density 7600 kg per m 3 . Determine the mass and polar moment of inertia of the propeller.

CHAPTER

3

Propeller Theory 3.1

Introduction

A study of the theory of propellers is important not only for understanding the fundamentals of propeller action but also because the theory provides results that are useful in the design of propellers. Thus, for exampl~, pro peller theory shows that even in ideal conditions there is an upper limit to the efficiency of a propeller, and that this efficiency decreases as the thrust loading on the propeller increases. The theory also shows that a propeller is most efficient if all its radial sections work at the same efficiency. Fi nally, propeller theory can be used to determine the detailed geometry of a propeller for optimum performance in given operating conditions. Although the screw propeller was used for ship propulsion from the be ginnIng of the 19th Century, the first propeller theories began to be devel oped only some fifty years later. These early theories followed two schools of thought. In the momentum theories as developed by Rankine, Green hill and R.E. Froude for example, the origin of the propeller thrust is ex __ plained entirely by the change .in the momentum of the fluid due to the .' propeller. The blade element theories, associated with Weissbach, Redten --) bacher, W. Froude, Drzewiecki and others, rest on observed facts rather than on mathematical principles, and explain the action of the propeller ill terms of the hydrodynamic forces experienced by the radial sections (blade elements) of which the propeller blades are composed. The momentum the ories are based on correct fundamental principles but give no indication of

28

l

Propeller Theory

29

the shape of the propeller. The blade element theories, on the other hand, explain the effect of propeller geometry on its performance but give the er roneous result that the ideal efficiency of a propeller is 100 percent. The divergence between the two groups of theories is explained by the circula tion theory (vortex theory) of propellers initially formulated by Prandtl and Betz '(1927) and then developed by a number of others 1;0 a stage where it is not only in agreement with experimental results but may also be used for the practical design of propellers.

3.2

I \

\

I

I1

Axial Momentum Theory

In the axial momentum theory, the propeller is regarded as an "actuator disc"which imparts a sudden increase in pressure to the fluid passing through it. The mechanism by which this pressure increase is obtained is ignored. Further, it is assumed that the resulting acceleration of the fluid and hence the thrust generated by the propeller are uniformly distributed over the disc, the flow is frictionless, there is no rotation of the fluid, and there is an un limited inflow of fluid to the propeller. 'The acceleration of the fluid involves a contraction of the fluid column passing through the propeller disc and, since this cannot take place suddenly, the acceleration takes place over some distance forward and some distance aft of the propeller disc. The pressure in the fluid decreases gradually as it approaches the disc, it is suddenly in creased at the disc, and it then gradually decreases as the fluid leaves the disc. Consider a propeller (actuator disc) of area Ao advancing into undis turbed fluid with a velocity VA' A uniform velocity equal and opposite to VA is imposed on this whole system, so that there is no change in the hydrody namic forces but one considers a stationary disc in a uniform flow of velocity VA, Let the pressures and velocities in the fluid column passing through the propeller disc be Po and VA far ahead, PI and "A + VI just ahead of the disc, P~ and VA +VI just behind the disc, and P2 and "A +V2 far behind the disc, as shov,!1 in Fig.3,1. From considerations of continuity, the velocity just ahead and just behind the disc must be equal, and since there is no rotation of the fluid, the pressure far behind the propeller must be equal to the pressure far ahead, Le. P2 = PO.

The mass of fluid flowing through the propeller disc per unit time is given by:

I

I

l~.


30 FLUID COLUMN

ACTUATOR DISC AREA . Ao

-r---' '-._.-.'-'_.-i=-'-'-'-'-'

FAR ASTERN

FAR AHEAD

I

PRESSURES

VELOCITIES

-p.' 1

PRESSURE VARIATION Figure 3.1 : Action of an Actuator Disc in the Axial Momentum Theory. .\

(3.1)

where p is the density of the fluid. This mass of fluid is accelerated from a' v.elocity VA to a velocity VA + V2 by the propeller, and since the propeller thrust T is equal to the change of axial momentum per unit time: (3.2)

. The total power delivered to the propeller PD is equal to the increase in the kinetic energy of the fluid per unit time, i.e. :

(3.3)

....

-.-_... -.-_._-_. __ ..._.... ,

.

Propeller Theory This delivered power. is also equal to the work done by the thrust fluid per unit, time, .j.e

the

011

l

•. :.

.'

:.',.

It therefore follows that:

(3.5)

, .

i.e. half the increase in axial velocity due to the propeller takes place ahead of it and half behind it. The same result may be obtained in a different way. By applying the Bernoulli theorem successively to the sections far ahead and ju.st ahead of the propeller, and to the sections far behind andjust behind the prop'eiler, one obtains: . . . . ,.

Po P2+

+ 2"1 PVA 2

= PI

t.

1

2" P(VA

+ Vl)

2

! (VA + V2)2 = P~.+ ,! p(~ + Vl)2 .

"

..

~o

.,

that,noting that P2 =, Po:

P~ - Pl

...•

= ! P [(VA + V2)2 -

~

. . ' ...!

.~.

.

,

2 VA j

= P (VA + ! V2) V2 ..

~

.

The propeller thruSt is given by:

so that by'comparing Eqns. (3.2) and (3.9), Ol~e again obtains Eq!1' (3:5). The useful work done by the propeller per unit time is TVA. The efficiency of the propeller is therefore: T]i

~-~,

TVA

1

PD

l+a

=-

_

_,-- --- ---

_-_

- '.- -~.

.

.

.(3.10)


32

where a= vI/VA is the axial inflow factor,· and VI and V2 are the axial induced velocities at the propeller and far behind it. The efficiency 'f}i is called the "ideal efficiency" because the only energy loss considered is the kinetic energy lost in the fluid column behind the propeller, i.e. in the propeller slipstream, and the other losses such as those due to viscosity, the rotation of the fluid and the creation of eddies are neglected. The thrust loading coefficient of a propeller is defined as:

T

CTL

Substituting the value of T from Eqn. (3.2) and noting that 2aVA, and a = (1/'f}i) - 1, one obtains:

V2

=

2 'f}i

(3.11)

= "21 P A 0 v:A 2

= 1 + \1'1 + CTL

VI

= a't-A, (3.12) \ i

This is an important result, for it shows that'themaximum efficiency of-a propeller even ullder ideal.conditions is limited to a Value less than 1, and that this efficiency decreases as the thrust loading increases. It therefore fol lows that for a given thrust, the larger the propeller the greater its efficiency, other things being equal. Example 1 \

A propeller of 2.0 m diameter produces a thrust of 30.0 leN when advancing at a speed of ;4.0 m per sec in sea water. Determine the power delivered to the propeller, the velocities in the slipstream at the propeller disc and at a section far astern, the thrust loading coefficient and the ideal efficiency.

D

== 2.0m

T == 30.0leN

Ao p

= ~ D 2 = 3.1416m

2

= 1025kgm- 3

so that: 1025 x 3.1416 (4.0 + vd 2 Vl = 30.0 x 1000

Propeller Theory

33

which gives: VI

a

= 0.9425ms- 1

=

V2

= 1.8850ms- 1

7Ji

1 - = =1+a

PD = CTL

1,

0.2356

=

TVA 7Ji

=

0.8093 30.0 X 4.0 0.8093

T ~pAOVA2

=

~

X

= 148.27kW 30.0 X 1000 1025 X 3.1416 X 4.02

=

1.1645

If GT L r~duces to zero, i.e. T = 0, the ideal efficiency TJi becomes equal to 1. If, on the other hand, "A tends to zero, TJi also tends to zero, although the propeller still produces thrust. The relation between thrust and delivered power at zero speed' of advance is of interest since this condition represents the practical situations of a tug applying static pull at a bollard or of a ship at a dock trial. For an actuator disc propeller, .the delivered power is given by:

a

(3.13) As 1(4. tends to zero, 1 + VI

+ GTL tends to VCTL'

= [ T3

2p A o

so that in the limit:

]~ //

that is,

(3.14)

.L


34

This relation between thrust and delivered power at zero velocity of ad vance for a propeller in ideal conditions thus has a value of y'2. In actual practice, the value 'of this relation is considerably less. Example 2 A propeller of 3.0 m diameter absorbs 700 kW in the static condition in sea water. What is its thrust? D = 3.0m p

=

PD = 700kW

Ao

1025kgm- 3

T 3 = 2p.4. o P'i> = 2 X 1025 x 7.0686

T

3.3

X

(700

X

1000)2 kgm- 3 m2 (Nms- 1 )2

= 7100.39 X 1012 N3 = 192.20kN

Momentum Theory Including Rotation

In \this theqry, also sometimes called the impulse theory, the propeller is regar:ded as imparting both a.."Cial and angular acceleration to the fluid flowing throlfgh the propeller disc. Consider a propeller of disc area Ao advancing into un,disturbed water with an axial velocity l-A. while revolving with an angular velocity w. Impose a uniform velocity equal and opposite to l-A on the whole system so that the propeller is revolving with an angular velocity w at a fixed position. Let the axial and angular velocities of the fluid then be VA + Vl and Wl at the propeller disc and VA + V2 and W2 far downstream, as shown in Fig. 3.2. The mass of fluid flowing per unit time through an annular element between the radii rand r + dr is' given by:

dm = pdAo (VA

+vd

where dAo is the area of the annular element.

(3.15)

Propeller Theory

35

FAR ASTERN

ACTUATOR DISC AREA A o ANGULAR VELOCITY w

FAR AHEAD

FLUID VELOCITIES

VA,+v2

AXIAL

W2

ANGULAR

VA+v,

w,

Figure 3.2: Action of a Propeller in the Impulse Theory.

. The thrust developed by the element is determined from the change in the axial momentum of the fluid per unit time: (3.16)

The torque of the element is similarly obtained from the change in angular momentum per unit time: (3.17)

The work done by the element thrust is equal to the increase in the axial kinetic energy of the fluid flowing through the annular element..Per unit time, this is given by:

that is,

~L

_

36


so that: (3.18)

This is the same result as obtained in the axial momentum theory, Eqn. (3.5). The work done per 1,1nit time by the element torque is simi~ larlyequal to the increase in the rotational kinetic energy of the fluid per unit time, i.e. :

dQWI -

~,dmr2 [w~

- 0]

- ~ p dAo("A + VI) W2 r 2 W2

-

~ dQ W2

so that, WI

- ~ W2

;'(3.19)

Thus, half the angular velocity of the fluid is acquired before it reaches the propeller and half after the fluid leaves the propeller. The total power expended by the element must be equal to the increase in the, total kinetic energy (axial and rotational) per unit time, or the work done by the element thrust and torque on the fluid passing through the element ,per unit time:

that is,

and the efficiency of the element is then:

(W - WI)"A ("A + VI) W

=

1- ~ W

l+~

= 1- a' l+a

(3.20)

Propeller Theory

37

where a' = wl/w and a = vl/VA are the rotational and axial inflow factors, VI and V2 are the axial, induced velocities at the propeller and far down stream, Wl and W2 being the corresponding angular induced velocities. It may be seen by comparing this expression for efficiency, Eqn. (3.20), with theexpression obtained in the axial momentum theory, Eqn. (3.10), that the effect of slipstream rotation is to reduce the efficiency by the factor (1- a') .. By making the substitutions: dAo - 21r r dr, Wl = a' w

a"A,

2a' w

Vl W2

-

in Eqns. (3.16) and (3.17), one obtains:

"

/' dT -

.

/dQ -

41rprdr"A 2 a(1+a)

(3.21)

41rpr 3 dr"ACLI'a'(1+a)

(3.22)

The efficiency of the annular element is!'then given by:

7]

=

dTVA dQw -

41r P r dr "A 2 a(l+ a)"A . a VA 2 41rpr 3 drVAwa'(1+a)w = a' w2 r 2

(3.23)

Comparing this with Eqn. (3.20), one then obtains:

or, a' (1 - a') w2 r 2

=

a (1 + a) VA 2

(3.24)

This gives the relation between the axial and rotational induced velocities in a propeller when friction is neglected. Example 3 A propeller of diameter 4.0 m has an rpm of 180 when advancing into sea water at a speed of 6.0 m per sec. The element of the propeller at 0.7 R produces a thrust of 200 kN per m. Determine the torque, the axial and rotational inflow factors, and the efficiency of the element.


38 D == 4.0m r

n == 180rpm

=

3.05- 1

VA

= 6.0ms- 1

dT == 200kNm- 1 dr

== 0.7R'== 0.7x2.0 = 104m

w = 211" n == 611" radians per sec

so that,

411" x 1025 x 1.4

X

6.0 2 a(l + a) = 200 x 1000

which gives, a

== 0.2470

a' (1 -a') w 2 r 2 == a (1 + a) VA 2

that is,

a'(l - a')(611")2 x 1.4 2 == 0.2470(1 + 0.2470) x 6.0 2 or,

a' == 0.01619

dQ == 471" P r 3 VA w a' (1 dr _ 471" x 1025

X

1.43 x 6.0

== 80.696 kN m m

1]

= ==

+ a) X

671" x 0.01619 x 1.2470

1

1 - 0.01619

1- a' == 1 + 0.2470 == 0.7889 l+a

dT VA

dr

~w

200 x 6.0 == 0.7889 X 671"

== 80.696

j

I

l·~l ,

'

.~

I

Propeller Theory

3.4

39

Blade Element Theory

The blade element theory, in contrast to the momentum theory, is concerned with how the propeller generates its thrust and how this thrust depends upon the shape of the propeller blades. A propeller blade is regarded as being com posed of a series of blade elements, each of which produces a hydrodynamic / force due to its motion through the fluid. The axial,component of this hydro dynamic force is the element thrust while the moment ~bout the propeller axis of the tangential component is the element torque. The integration of the element thrust and torque over the radius for all the blades gives the total thrust and torque of the propeller. L

o

1 s

~J

~c---J

Figure 3.3: Lift and Drag of a Wing.

Consider a wing of chord (width) c and span (length) s at an an gle of attack Q to an incident flow of velocity V in a fluid of den sity p, as shown in Fig. 3.3. The wing develops a hydrodynamic force whose components normal and parallel to, V are the lift L and the drag D. One defines non-dimensional lift and drag coefficients as follows:

J___


40

L ~pAV2

CD

=

(3.25)

D ~pAV2

.

,

where A = s c is the area of the wing plan form. These coefficients depend upon the shape of the wing section, the aspect ratio sic and the angle of attack, and are often determined experimentally in a wind tunnel. These experimental yalues may then be used in the blade element theory, which may thus be said to rest on: observed fact.

(0) WITHOUT INDUCED VELOCITIES

\.

dD

~

_ _-2nnr - - -

(b) WITH INDUCED VELOCITIES

Figure 3.4 : Blade Element Velocities and Forces.

Now consider a propeller with Z blades, diameter D and pitch ratio PI D advancing into undisturbed water with a velocity VA while turning at a rev olution rate n, The blade element between the radii rand r + dr when expanded will have an incident flow whose axial and tangential vel<:Jcity components are VA and 211" n r respectively, giving a resultant velocity VR

"-Jr--

J.-!l!t

Propeller Tbeory

/.

~

/~

~

c

f<

41

at an angle of attack a, ~ shown in Fig. 3.4(a). The blade element will then produce a lift d!,:-and a drag eyh where:

(3.26)

If the thrust and torque produced by the elements between rand r for all the Z blades are dT and dQ, then from Fig. 3.4(a);···-·' 1 j Z dT = dLcos f3 - dD sinf3 = dLcisf3

(

dD ) 1- dL tanf3

@

dQ = dL sin{3 f dD cos {3 = ;ll/cos {3 (tan (3 +

where I

+ dr

~~)

(3.27)

VA

tan{3 = ' - ,21rnr

•

Putting tan')' = dDjdL, and writing dL and dD in terms of CL and CD, one obtains: dT -

Z GL' ~ pcdr VJ cosf3 (1- tanf3 tan')')

dQ -

r Z CL

.

~ p;d~ VA cos f3 (tan f3+ tan')')

(3.28)

The efficiency of the blade element is then:

@ ""

7J = dQ21r

=

1 - tan (3 tan ~ tan{3 21rnr tan{3 + tan')' :::: tan(f3 + ')')

(3.29)

It "'ill be sho~n later that for a propeller to have the maximum efficiency in given conditions, all its blade elements must have the same efficiency. Eqn. (3.29) thus also gives the efficiency of the most efficient propeller for the specified operating conditions.

../ I f the propeller works in ideal conditions, there is no drag and hence tan..... = 0, resulting in the blade element efficiency and hence the efficiency


42

j

of the most efficient propeller being 1] = 1. This is at variance with the esuIts of the momentum theory which indicates that if a propeller produces a thrust greater than zero, its efficiency even in ideal conditions must be less than 1.

J

The primary reason for this discrepancy lies in the neglect of the induced velocities, i.e. the inflow factors a, a'. If the induced velocities are (;ak~ ac~, as shown in Fig. 3.4(b), one obtains:

dT = ZCL·~pcdrV~cos!h (l-tan,Bltan,) dQ

= r Z C L . ~ P c dr V~ cos (3I

(3.30) (tan,B[

+ tan,)

and: 1]

=

dTVA· dQ2ran

-

VA 1'- tan,BI tan, = 271" n r tan,B[ + tan,

tan,B

1 - a' tan,B[ tan,B tan ,B[ =--. = tan,B[ tan (,B[ + ,) 1 + a tan (,BI + ,)

.(3.31)

since, tan,B =

VA 271" n r

and

tan,B[

"A (1 + a) = 271"nr(1-a')

l+a = tan,B 1 - a'

In Eql1. (3.31), the expression for efficiency consists of three factors: (i) 1/(1 t a), which is associated with the axial induced velocity, Eqn. (3.10), (ii) (1 - a'), which reflects the loss due to the rotation of the slipstream, and (iii) tan,BrI tan(,B[ + ,), which indicates the effect of blade element drag. If there is no drag and tan, = 0, the expression for efficiency, Eqn. (3.31), becomes identical to the expression obtained from the impulse theory, Eqn. (3.20). In order to make practical use of the blade element theory, it is necessary to know CL, CD, a and a' for blade elements at different radii so that dT/dr and dQ/dr can be determined and integrated with respect to the radius r. C L and CD may be obtained from experimental data, and a and a' with the help of the momentum theory. Unfortunately, this procedure does not y~eld realistic results because it neglects a number of factors.

Propeller Theory

43

Example 4 A four bladed propeller of 3.0 m diameter and 1.0 constant pitch ratio has a speed of advance of 4.0 m per sec when running at 120 rpm. The blade section at 0.7R has a chord of 0.5 m, a no-lift angle of 2 degrees, a lift-drag ratio of 30 and a lift coefficient that increases at the rate of 6.0 per radian for small angles of attack. Determine the thrust, torque and efficiency of the blade element at 0.7R (a) neglecting the induced velocities and (b) given that the axial and rotational inflow factors are 0.2000 and 0.0225 respectively.

Z=4

D

=

3.0m

P D

=

1.0

n = 120rpm = 2.0 s-l

x=

r

Ii =

30

p ~' 1025kgm- 3

= 6.0 per radian

(a)

=

c = 0.5m

0.7

Neglecting induced velocities:

=

tanlp = P/D 71" X tan!,

=

VA 271"nr

tan...,'

=

CD CL

o·

1.0 71" x 0.7

=

271"

= ~ = 30

0.4547

If> = 24.4526°

4.0 2.0 x (0.7 x 1.5)

X

=

0.3032

0.03333

8CL 80: (0:0+0:)

~r2

=

~2

=

190.0998 m2 S-2

+ (271"nr)2 =

dT = Z CL dr

.J._ _

=

6.0 2 + 7.5878 = 1.0040 180/71" 4.0 2 + (271"

4pc vJ cos {3 (1 -

f3

= 16.8648°

'Y = 1.9091°

= If> - {3 = 7.5878°

CL = R

=

X

2.0

X

tan{3 tan 1')

1.05)2

44

Basic Ship Propulsion dQ dr = r Z C L ~ P c V~ cos f3 (tan f3 + tan')')

Substituting the numerical values calculated:

dT dr 1]

(b)

= 185.333 k.l~ m- 1 =

dQ = 66.148kNmm:"1 dr

tanf3 = 0.8918 tan(3+:)

a = 0.2000

Given:

v~

:::

[(1

+ a)

. a' ~ 0.0225

\'A]2 + [(1-a') 271"nr]2

::: [(1 + 0.2000) 4.0]2 + [(1 - 0.0225) 271" x 2.0

X

1.05]2

= 23.0400 + 166.3535 ::: 189.3935 m 2 s-2 tanf3I =

PI

~... (1 + a) 4.0 (1 + 0.2000) = 0.3722 = 271"nr(1-a') 271" x 2.0 x 1.05 (1 - 0.0225)

= 20.4131 0

a: = !p - f3I = 24.4526 - 20.4131 = 4.0395

6. 0

2+ 4.0395 = 0.6325 180 7r

Substituting these values in:

dT

dr = Z CL ~ pcV~ cosf3r (1- tanf3r tan')')

dQ 1 2 dr = 7'ZCL 2 PcVRcosf3I (tanth+tan')')

,

-f$:

0

45

Propeller Theory one obtains:

dT dr

=

113.640kNm- 1

dQ

dr

=

48.991 kNmm- 1 1-

1]

3.5

a'

= 1 +a

tan(3r tan((3r + "Y)

=

1- 0.0225 0.3722 x-1 + 0.2000 0.4104

= 0.7383

Circulation Theory

The circulation theory or vortex theory provides a more satisfactory explanation of the hydrodynamics of propeller action than the momentum and blade element theories. The lift produced by each propeller blade is ~xplained in terms of the circulation arou;nd it in a manner analogous to the lift produced by an aircraft wing, as described in the following. . : I ~

~ ::.V+V ..

.~

=::::..v-v~

: & :-v~

vr=k

(0) VORTEX FLOW

I

(b) UNIFORM FLOW

i

(c) VORTEX IN UNIFORM FLOW

Figure 3.5: Flow olan Ideal Fluid around a Circular Cylinder.

Consider a flow in which the fluid particles move in circular paths such that the velocity is inversely proportional to the radius of the circle, Fig.3.5(a). Such a flow is called a vortex flow, and the axis about which the fluid particles move in a three dimensional flow is called a vortex line. In an ideal fluid, a vortex line cannot end abruptly inside the fluid but must either form a closed curve or end on the boundary of the fluid (Helmholz theorem). A circular cylinder placed in a uniform flow of an ideal fluid, Fig. 3.5(b), will experience

1___

46


no force because of the symmetry of the velocity and pressure distributions around the cylinder (D'Alembert's paradox). If, however, a vortex flow is superposed on the uniform flow, there will be an asymmetry in the flow, the resultant velocity will increase and the pressure decrease on one side of the cylinder as compared to the other, resulting in a force (lift) normal to the direction of the uniform flow, Fig.3.5(c).

I'

The line integral of the velocity along a closed curve around the cylinder . is called circulation. If the cylinder is placed in a uniform flow of velocity V on which is superposed a vortex flow such that the velocity tangential to a circle of radius r is given by v = k/r, then the circulation r obtained by taking the line integral around the circle of radius r is:

f v ds = 1 ~ r dB = 21r

r=

'27r k

(3.32)

The contribution due to the uniform velocity V is zero, the contribution

on one side of the cylinder cancelling that on the other. The circulation is

independent of r, and it can be shown that the same value is obtained for

any closed curve around the cylinder by transforming the curve into radial

and tangential segments.

The flow past an aerofoil can be regarded as composed of a uniform flow

of velocity V and a vortex flow of circulation r, the resulting asymmetry in

the floW: causing the aerofoil to develop a lift. This lift per unit length (span)

of the aerofoil is given by the Kutta-Joukowski theorem:

L = pry

(3.33)

where the circulation r depends upon the shape of the aerofoil and its angle

of attack. A simple but non-rigorous proof of this result can be obtained as

follows.

Referring to Fig. 3.6, let th.e tangential velocity due to the circulation be v. If the pressure on the upper and lower surfaces of the aerofoil at a section

a distance x from the leading edge are Pu and PI, then by the Bernoulli

theorem:

Pu + p (V + v)2 = PI + p (V - v)2 so that the difference between the pressures on the lower and upper surfaces of the aerofoil at the section is: '

!

!

6. P

=

PI - Pu

=

2p V v

'"

I

47

Propeller Tbeory v

x __-\--- ...:;.

-'1.-.10

_~~~x

'---_dX

y

......

t

.... ..--- V

v

Figure 3.6: Circulation around an Aerofoil.

If the x-coordinate of the trailing edge lis aerofoil per unit length (span) is:

L

Xl,

the lift produced by the

=

l·['t>.PdX = pV['2VdX = pLC~dX+ J.:-VdX] V

-

pry

as given in Eqn. (3.33).

If a wing of infinite aspect ratio with an aerofoil cross-section is given a velocity V, the flow around it is initially as shown in Fig.3.7(a), with a stagnation point 51 near the leading edge and a stagnation point 52 upstream of the trailing edge on the upper surface with flow taking place around the trailing edge towards 52 in an adverse pressure gradient. Such a flow is unstable and as a result a vortex is shed from the trailing edge causing the stagnation point 52 to move to the trailing edge and a circulation to develop around the wing, Fig. 3.7(b). The vortex, which is shed from the aerofoil at the start of the flow is called the starting vortex while the vortex associated with the circulation around the wing is called the bound vortex. In a wing of finite span, the starting vortex land the bound vortex cannot end abruptly in the fluid and there must exist vortex lines which connect the

,

lc_


48 STARTING

VORTEX

:;;

----

.

(0)

(b)

Figure 3.7: Start of Flow past an Aerofoil.

mAILING VORTEX

(0) . VORTEX SYSTEM OF A .

W1NG AT START

(b) HORSESHOE VORTEX CONFIGURATlON

\

Figure 3.8: Vortex System of a Wing of Filiite Span.

ends of the starting vortex and the bound vortex, as shown in Fig.3.8(a). These vortex lines which are shed downstream from the wing tips are called trailing vortices. These trailing vortices, which are aligned parallel to the velocity V in contrast to the bound vortex which is at right angles to V, belong to the class of free vortices which do not produce lifting forces. In the course of time, the starting vortex is left far downstream of the wing, and the vortex system has a horseshoe shape, Fig.3.8(b). The trailing vortices induce a dow:nward velocity in the flow behind the wing. The magnitude of this velocity far behind the wing is twice that at the wing because the trailing vortices extend infinitely in both directions far behind the wing while at the wing the trailing vortices extend infinitely only in one direction.

Propeller Theory

49

In the vortex system shown in Fig. 3.8, the circulation along the span of the wing has been taken to be constant. Actually, however, the circulation in a wing of finite span decreases from a maximum at mid-span to zero at the ends, and this is represented by the vortex system shown in Fig. 3.9. The trailing vortices are shed not only from the wing tips but from all along

Figure 3.9: Vortex System of a Wing with Circulation Varying along the Span.

\

the trailing eage, forming a vortex sheet. The strength of a free vortex shed from any element along the trailing edge is equal to the change in ~irculation_ across that element. The vortex system of a propeller is similar to that of a wing as described

' '----In the foregoing. Each blade of the propeller is represented by a bound vortex or lifting line of strength varying along the length of the blade, and a vortex sheet is shed from the trailing edge. Since the propeller revolves about its axis while simultaneously advancing along it, this trailing vortex sheet is helicoidal in shape, and there are as many such vortex sheets as there are blades in the propeller. These trailing vortex sheets produce induced velocities that are perpendicular to the vortex sheets, the induced velocity at the blade being half the induced velocity far downstream. Consider a propeller with Z blades and diameter D advancing in an inviscid fluid with a velocity VA while turning at a revolution rate n. Let the circulation at a radius r of the blade be f, and let the induced velocity far downstream be u with axial and tangential components U a and Ut. A

~---

.

50

Basic Ship Propulsion TRAILING VORTEX AT RADIUS r U

Q

BOUND

TRAILING VORTEX SHEET

.1/ VORTEX

r \ '~-==/:.~=iW-\:t=\==~~==l

l' ----~--.r--~A 1. n FAR AHEAD

FAR

ASTERN Z

I

=

4

!

V

.cbr I I I

cPr i r

Ut

9.

0

2JTr

I

cbr

v

I

\

Figure 3.10: Vortex System of a Propeller.

relationship between the circulation r and the induced velocity Ut is found by taking the line integral of the velocity along the closed curve defined by the ends of a cylinder of radius r. extending from far ahead of the propeller to far behind it, the end circles being connected by two parallel straight lines very close to each other, as shown in Fig. 3.10. The velocity along the circle far ahead is zero and along the circle far astern is Ut. The line integrals along the two parallel straight lines cancel each other, so that the circulation is obtained as:

zr

27r rUt

(3.34)

:

Propeller Theory

51

provided that there are so many blades Z that Ut is constant along the circle far astern. The effect of a finite number of blades is considered later.

VA '_1 dQ

rZ

l4--~I::.--_--->~ ~-~----+-J1 __

i

~L

~_ _"'--_ _ 2 n n r ~

.~~ . t u t

Figure 3.11 :Velocities and Forces on a Blade.Element in Inviscid Flow.

Fig. 3.11 shows the velocities and forces acting on a blade element of length I dr at the radius r. The velocities are tak~n relative to the blade element. The induced velocities at the blade element are half those far behind the propeller, and since the fluid is assumed to be without viscosity there is no drag. The lift on the blade element is dL. The ideal thrust and torque due to the elements at radius r for all the blades are denoted by dT;. and dQi so that the axial and tangential force components on each blade element are dT;,/Z and dQi/rZ as shown in the figure. VR is the resultant velocity, Q the angle of attack, and {3 and {3I are the hydrodynamic pitch angles excluding and including the induced velocities respectively.. The lift on a blade element of length dr at the radius r is now obtained by the Kutta-Joukowski theorem: 27frUt

p--

Z

substituting for

r

V' d R

r

from Eqn. (3.34). From Fig. 3.11, one obtains:

z1 dTi

= dL cos {3I

1 -z dQi

= dL sin {3I

7'

(3.35)


52

so that, using Eqn. (3.35):

dTi

- 27l" pUt V R cos (3r r dr

dQi

-

21rpUt VR

(3.36)

filin(3rr 2 dr

and: I

77i

-

VA cos (3r dTiVA = ---21r n r sin (3] dQi 21r nr

tan (3 tan (3]

(3.37)

where 77~ is the ideal efficiency of the blade section at radius r. One may now derive the condition for a propeller of maximum efficiency, or the "minimum energy loss condition"derived by Betz (1927). Suppose that there are two radii rl and r2 where the blade element efficiencies are 77h and 77b, with 77h being greater than 77h. It is now possible to modify the design of the propeller (by changing, the radial distribution of pit~h, for example) in such a way that the torque is increased by a small amount at the radius rl and decreased by an equal amount at the radius r2' However, because 77h is greater than 77i2' the increase in thrust at rl will be greater in magnitude than the decrease in thrust at r2. There will thus be a net increase in the total thrust T i without any increase in the total torque Qi of th~ propeller, and hence an increase in its efficiency. This process of increasing the efficiency of the propeller can be continued so long as there exist two radii in the propeller where the blade element efficiencies are not equal. If the efficiencies of the blade elements at all the radii from root to tip are equal, then the efficiency of the propeller cannot be increased further, and one has a propeller of the highest efficiency or minimum energy loss. The condition for minimum energy loss is therefore that 77;1 be independent of r, i.e. for all radii: tan (3 1 V;t tan(31 = - - = - - 77i 77i 21r n r

(3.38)

where l]i = 77; aild is the ideal efficiency of the propeller. Eqn. (3.38) implies that the vortex sheets shed by the propeller blades have the form ofhelicoidal surfaces of constant pitch VA !(77i n).

I

Jt

!

!

I

Propeller Theory

53

From the blade section velocity diagram in Fig. 3.11, it can be shown that: VR VA

=

cos (f3J - (3) sin{3

=

tan{3J - tan{3 tan{3 (1 + tan2 (31)

=

tan{3J (tan{31 - tan(3) tanf3 (1 + tan 2 (31)

1

2 Ua VA

!

Ut

VA

(3.39)

The ideal thrust loading coefficient is defined as:

o ._

11

TLt -

1

2P

A

0

v:A 2

where Ao = 1r D2 /4 is the disc area of the propeller. The ideal thrust loading coefficient for a blade element at radius r is then:

=

27r pUt V n cosf31 rdr

EpD

2

VA 2

16 Ut VR cos f31 r dr

D2 "A 2

(3.40)

Using Eqns. (3.38), (3.39) and (3.40), one can write:

\

dCTLi

~

so that: CTLi =

I-I

= I(A,TJi, r)

(3.41)

(A, 1]i, r)dr = F(A,1]i)

where A = VA /1rnD = r tanf3/R. The function F(A,fJi) was calculated by Kramer (1939) for different values of Z, the number of blades. His results are usually given in the form of a diagram. The Kramer diagram, which often forms the starting point for designing a propeller using the circulation theory, is given later. In determining the circulation r around a propeller blade at radius r, Eqn. (3.34), it has been assumed that the tangential induced velocity Ut is

J.

l

.\\

r r~~"b

~

!*'jUI"IlI,lt

_

54


constant along the circular path in the slipstream far behind the propeller. This implies that the propeller has an infinite number of blades. The effect of a finite number of blades was calculated by Goldstein (1929) for a propeller having an optimum distribution of circulation. The effect of a finite number of blades is taken into account by incorporating a correction factor, the Goldstein factor t\, , in Eqn. (3.34) and the subsequent equations based on it. Goldstein factors have been calculated and are given in the form of diagrams with t\, as a function of tan Ih and x = r / R for different values of Z. The use of Goldstein factors to account for the finite number of blades in a propeller is valid only for a lightly loaded propeller operating in a uniform velocity field and having an optimum distribution of circulation. When these conditions are not fulfilied, it is more correct to use the induction factors calculated by Lerbs (1952). However, the Goldstein factors are much simpler to use than the Lerbs induction factors. Many propeller design methods based on the circulation theory therefore use the Goldstein factors. In order to use the circulation theory for propeller design, it is necessary to be able to calculate the lift coefficient required at each blade section of the propeller. By definition, the lift coefficient for the blade section 'at radius r is given by:

(3.42)

Substituting for dL from Eqn. (3.34) after incorporating the Goldstein fac tor, one obtains:

so that: CL

~

=

~x

t\,

sinlh tan(/h - f3)

(3.43)

where c is the chord length of the blade section at radius r I x = r / R, and use has been made of the Eqns. (3.39). Eqn. (3.43) is used to determine the detailed geometry of the blade sections when designing a propeller. The use of the circulation theory in propeller design is considered in Chapter 9.

Propeller Theory .

55

Example 5 A propeller of 5.0 m diameter has an rpm of 120 and a speed of advance of 6.0 m per sec when operating with minimum energy loss at an ideal efficiency of 0.750. The root section is at 0.2 R. Determine the thrust and torque of the propeller in an inviscid fluid neglecting the effect of a finite number of blades.

D = 5.0m p

7];

r

= 1025kgm-3

Root section x == -

=

tan{J tan{J[

= 0.750 =

7]i

tan{J

=

R

7

=

6.0

0.190986 x

tan{J 0.750

~Ut = tan{J[ (tan{J[ - tan{J) VA

= 0.2

-----= 7r X 2.0 x x x 5.0

=

tan{J = tan{J[

= 120rpm = 2.0s- 1

n

tan {J (1 + tan2 (3[)

-'\1: cos({J[ - (J)

V

R -

sin{J

A

dT, dr

The subsequent calculation is carried out in a tabular form, using Simpson's rule for the integration.

x

r

tan{J

tan{J[

~utlVA

m 0.2 0.3 0.4 0.5 0.6 0.7 0.8

J..__

0.500 0.750 1.000 1.250 1.500 1.750 2.000

0.9549 0.6366 0.4775 0.3820 0.3183 0.2728 0.2387

1.2732 0.8488 0.6366 0.5093 0.4244 0.3638 0.3183

0.1619 0.1645 0.1510 0.1348 0.1199 0.1071 0.0963'

Ut

{3[o

{J0

51.8540 40.3255 32.4816 k6.9896 22.9970 19.9905 17.6568

43.6793 32.4816 25.5228 20.9055 17.6568 15.2610 13.4270

ms- 1 1.9430 1.9734 1.8121 1.6176 1.4385 1.285~

1.1561

~r

it


56 x

tanf3

r

tanf3I

~utlVA

Ut

m 0·9 1.0

x 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.::n22 0.1910

2.250 2.500

VR

ms- 1

dTi/dr kNm- 1

8.5996 11.0680 13.8228 16.7200 19.6956 22.7172 25.7687 28.8399 31.9248

33.2342 80.4315 136.0798 194.0184 251.9514 309.1963 365.6541 421.3677 476.4799

Ii

= '13 x

Qi = \,

0.2829 0.2546

'13 x

f3ro

f3 0

15.7984 14.2866

11.9808 10.8125

ms- 1 1.0479 0.9566

0.0873 0.0797

d,Qi/dr kNmm- 1

8M

21.1576 51.2044 86.6308 123.5160 160.3973 196.8405 232.7829 268.2506 303.3362

1 4 2 4 2 4 2 4 1

(0.1 x 2.5) x 6037.1403

f(Ti ) kNm- 1

f(Qi)

33.2342 321.7260 272.1596 776.0736 503.9028 1236.7852 731.3082 1685.4708 476.4799 6037.1403

21.1576 204.8176 173.2616 494.0640 320.7946 787.3620 465.5658 1073.0024 303.3362 3843.3618

kNmm- 1

= 503.095 kN

(0.1 x 2.5) x 3843.3618 = 320.280kNm

=

27r

503.095 x 6.0 = 0.7500 ::: 1J~ X 2.0 x 320.280

3.6 Further Development of the

0irculation Theory The circulation theory as discussed in the previous section has been used for designing propellers for over sixty years, although it has been necessary to use empirical or semi-theoretical corrections to bring the results of the theory in line with experiment, and to account for some of the simplifi cations in the theory. One of the major defects of the circulation theory

'1,

.'

\

t

I'I•.

1

iiI

Propeller Theory

57

discussed in Sec. 3.5 is that the propeller blade is represented by a vortex line or lifting line. This neglects the effect of the finite width of the blade on the flow around it due to which the induced velocity varies across the "width of the blade. The variation of the induced velocity along the chord of a blade section causes a curvature of the flow over the blade resulting in changes to the effective camber of the blade sections and the ideal angle of - attack." Corrections to account for the width of propeller blades were first calculated by Ludwieg and Ginzel (19"44) and used extensively thereafter. The Ludwieg-Ginzel curvature correction factors have now been superseded by more accurate lifting surface corrections such as those due to Morgan, Silm·ic and Denny (1968). The modern development of the circulation theory is based on considering the propeller blades as lifting"surf
58

I


fluid flows, i.e. by RANS (Reynolds Averaged Navier Stokes) solvers. The propeller parameters are calculated for inviscid flow and then used with the viscous flow equations in an iterative procedure to account for the interac tion between the inflow velocity and the velocities induced by the propeller. However, RANS solver approaches do not model the actual geometry of the propeller and the mutual interference between the propeller blade,s. It is believed that RA.~S solvers that' include wake calculations and model tur bulence properly will eventually prove better than surface panel methods because RANS solvers would also model the trailing vortex field of the pro peller more accurately.

Problems .,'

1.

A propeller of diameter 4.0 m operates at a depth of 4.5 m below the surface of water with a speed of advance of 5.0 m per sec and produces a thrust of 200 kN. Determine th~ pressures and velocities (with 'respect to the propeller) in the water far ahead, just ahead, just behind and far behind .the propeller on the basis of the axial momentum theory.

2. A propeller of diameter 3.0 m has a speed of advance of 4.5 m per sec. The velocity of water relative to the propeller in the slipstream far astern is 7.5 m per sec. Determine the thrust and the efficiency of the propeller using the axiai momentum theory. 3. A propeller is required to produce a thrust of 150 kN at a speed of advance of \ 6.0m per sec with an ideal efficiency of 0.7. What should be the diameter of the propeller on the basis of the axial momentum theory? \ '

4.

A propeller

of diameter 3.0 m is required to produce a thrust of 100 kN at zero speed of advance. Determine the power input required, using the axial :momentum theory.

5. A propeller of diameter 4.0 m has a speed of advance of 8.0 m per sec and an rpm of 150. The axial velocity of water with respect to the propeller is constant over the cross-section in the ultimate wake (i.e. far downstream) and has a value of 10.0 m per sec. Using the momentum theory considering both axial and angular momentum, determine the thrust and torque distribution over the propeller radius J and hence the total thrust and torque of the propeller and its efficiency. Consider only the radii from 0.2R to LOR at intervals of O.lR. 6.

A propeller of 5.0 m diameter has a speed of advance of6.0 m per sec and a revolution rate of 120 rpm. The root section is at 0.2R. The propeller operates

•i i

\

PropeIIer Theory

59

in an inviscid fluid of density 1025 kg per m3 and has an efficiency of 0.750. Determine the axial and rotational inflow factors a and a' at the different radii, and calculate the propeller thrust and torque if the propeller operates at the highest efficiency.

A four-bladed propeller of diameter 5.0 m and constant face pitch ratio 0.800 is advancing through undisturbed water at a speed of 6.0 m per sec and 120 rpm. The blades are composed ofradial sections all of which have a chord of 1.0 m, a lift coefficient of 0.1097 per degree angle of attack (measured from the no lift line), a no-lift angle of 2.0 degrees and a lift-drag ratio of 40. The root section is at 0.2R. Calculate the thrust, torque and efficiency of the propeller neglecting the induced velocities.

7.

8. The lift coefficients and hydrodynamic pitch angles of the blade sections of a screw propeller are as follows:

x = r/R CL

!3[

deg

0.2 0.345 52.0

0.4 0.292 32.6

0.6 0.224 23.1

0.8 0.183 17.7

0.9 0.165 15.9

Assuming that the lift contributions of the angle of attack and of the camber are equal, compute the angles of attack, camber ratios and pitch ratios at the given radii. Lift coefficient due to angle of attack is 0.1097 per degree, and the section camber ratio is 0.055 per unit lift coefficient.

A four-bladed propeller of 6.0 m diameter produces an ideal thrust of 500 kN at a speed of advance of 8.0 m per sec and 90 rpm. Determine the ideal thrust loading coefficient CTLi and the advance ratio >.. If for these values of CTLi and>. the ideal efficiency 'T/i is 0.815 and the Goldstein factors K. are as follows:

9.

x = r/R K.

0.2 1.141

0.3 0.4 1.022 . 0.976

0.5 0.943

0.6 0.926

0.7 0.861

0.8 0.765

0.9 0.595

1.0

determine the distribution of thrust and torque over the propeller radius and the ideal delivered power. A three-bladed propeller of diameter 6.2m has a speed of advance of 18.707 knots at 132 rpm and produces an ideal thrust of 1024.6kN. The ideal efficiency is 0.795 and the Goldstein factors are:

10.

x K.

= r/R:

0.2 1.104

0.3 0.980

Determine the values of

0.4 0.941 CL

0.5 0.917

0.6 0.890

~ at these radii.

0.7 0.834

0.8 0.739

0.9 0.569

1.0

CHAPTER

4

The Propeller in "Open" Water

4.1

Introduction

A propeller is normally fitted to the stern of a ship where it operates in water that has been disturbed by the ship as it moves ahead. The performance of the propeller is thus affected by the ship to which it is fitted, so that the same propeller will perform slightly differe~tly "behind" different. ships. If therefore one wishes .to determine the intrinsic performance characteristics of a propeller, unaffected by the ship to which it is fitted, it is necessary to make the propeller operate in undisturbed or "open" water. The performance chamcteristics of a propeller usually refer to the variation of its thrust, torque and efficiency with its speed of advance and revolution rate in open water. I

It is difficult to determine the characteristic,? of a full-size propeller either in "open" water or "behind" the ship by varying the speed of advance and the revolution rate over a range and measuring the thrust and torque of the propeller. Therefore, recourse is had to experiments with models of the propeller and the ship in which the thrust and torque of the model propeller can be conveniently measured over a range of speed of advance and revolution rate. However, before one embarks upon model experiments, it is essential to know the conditions under which the quantities measured on the model can be applied to the full-size ship. These conditions are obtained from the "laws of similarity" between a model and its prototype.

60

61


4.2

Laws of Similarity

The laws of similarity provide the conditions under which a model must be made to operate so that its performance will faithfully reflect the per~ formance of the prototype. For studying the performance of a propeller, . or indeed studying hydrodynamic phenomena in general through the use of models, the conditions that need to be fulfilled are: , (i) condition of geometrical similarity

(ii) condition of kinematic similarity (iii) condition of kinetic (dynamic) similarity

• I

\

. The first condition requires that the model be geometrically similar to the full-size body. For this, it is necessary that the ratio of every linear dimension of the body bear a constant ratio to the ,corresponding dimension of the model. Thus, if the diameter of a ship propeller is D S = 5 m and the diameter of the model propeller is D M 20 em, the scale ratio A = Ds / DM 25, and this should be held constant for all linear dimensions such as the boss diameter or the blade section chord lengths and thicknesses at corresponding radii.

=

=

. Kinematic similarity requires that the ratio of any velocity in the flow field of the full-size body to the corresponding velocity in the model be constant. This, in effect, means that the flow fields around the full-size body and the model are geometrically similar since the ratio of the velocity components at corresponding points are equal and hence the directions of the resultant velocities at these points are identical. For propellers, the condition for kinematic similarity can be conveniently expressed by considering the speed of advance and the tangential velocity at the blade tip: VAS VAM

7f

ns Ds

= 7fnM D M ,

or

VAS nsDs

=

VAM nMDM

that is, Js

J..

_

=

JM

(4.1)

62


thf.s ~ S and Ai referring to the ship and the model respectively.

J =\ VA/ry..D is called the advance coefficient.

r"-

.....,..~.....-/t

\,yI'The condition for kinetic or dynamic similarity is somewhat more compli , cated, and requires that the ratios of the various forces acting on the full-size body be equal to the corresponding ratios in the model. Consider a body in a flow in which the dimensions of length and time are'denoted by It. and T. Then, some of the forces acting on the body can be expressed in terms of l:. and T as follows: Inertia Force = mass

X

acceleration

Gravity Force = mass

X

gravitational acceleration

ex

PIt.3

X

g =

pgIt.3

Viscous Force = coefficient of viscosity x velocity gradient x

are~

L ex

f-L x

.~

Pressure Force = pressure

X

L 2 - f-L l:.2 T- 1

X

area

\

In this,' p and J.L are the density and coefficient of dynamic viscosity of the

fl~id, g is the acceleration due to gravity, and p th;p~ess~~~~·"····--~·'"·-Dynamic similarity then involves the following ratios: Inertia Force p}d4 T.- 2. }d2 T.- 2 V2 Gravity Force pgL 3 = gl:. = gL Inertia Force Viscous Force

pL4 T- 2 f-L l:.2 T- 1

Pressure Force Inertia Force

p l:.2 Pl:.4 T- 2

=

p (l:.2 T- 1 ) J.L

=

VL

IF P

VL lJ

p P = 1 )-2 pV2 (LTp -

J . .;;-"1 >;. .

Tbe Propeller in "Open" Water

63

In the foregoing, the various constants of proportionality have been taken as 1, and L, V and p are a characteristic length, a characteristic veloc ity and a characteristic pressure associated with the body and the flow around it, while v = pip is the kinematic viscosity of the fluid._The ratio of inertia force to gravity rorce;o-;:'r~ther its;quare root, is called the Proude number F n after' William Froude, who was among the first to show the connection between the gravity waves generated by a ship and its speed. The ratio of inertia force to viscous force is called the R~ynolds number Rn after Osborne Reynolds who studied the flow of viscous fluids. The ratio of pressure force to inertia force, expressed in the form pi! p V2, is called the pressure coefficient or sometimes the Euler number En, after Leonhard Euler, the famous 18th century mathematician; a special form of the pressure coefficient is called the cavitation number. The Froude number, the Reynoids number and the Euler number (or pressure coefficient) being ratios of forces are dimensionless numbers and have the same values in any. consistent system of units. . The condition for dynamic similarity requires that these force ratios for the full-size body be equal to the corresponding ratios'for the model, Le.:

=

V

.jgL'

Rn=

VL v

=

p ~ pV2

(4.2)

should be the same for both the full-size body and the model. If forces other than those considered here are involved, then other force ratios must be considered. Thus, if surface tension is important the Weber number W n = V 2 LII"\" where I"\, is the kinematic capillarity (surface tension per unit length I density) of the fluid, mm,;t be the same for the body and its model. In considering the dynamic similarity of propellers, one may take the char acteristic linear dimension as the propeller diameter D, the characteristic velocity as the speed of advance VA and the characteristic pressure as the static pressure at the centre of the propeller po: The condition for dynamic similarity then becomes:

J.. I

_


64

. / IV V'

/

F ns

=

FnM,

Le.

Rns

= RnM,

Le.

Ens

=

VAS

.;gI5S

=

VAM

.;gIJM

VAsDS = Vs

VAMDM

Pos V2 'iPS AS

POM V2 'iPM AM

VM

(4.3)

l

EnM,· Le.

I

=

I

Sometimes, other parameters instead of D, VA and PO are chosen for defining Fn , Rn and En' the choice depending upon the purpose for which these dimensionless numbers are required. Thus, for example, when considering the nature of the viscous flow around a propeller, the character istic length may be the length (chord) of the blade section at O.7R and the characteristic velocity the result~'nt of the axial and tangential velocities at the section (neglecting induced velocities), so that the Reynolds number of the propeller may be defined as:

RnO.7~ =

VO.7R CO.7R v

(4.4)

where: --'~"''''~''''''

!

/

;VO.7R \

4.3 ,

\

..•. "-~,,-,,

9J.'7n;.::.<'the chord lengthaf,~he blade section

atO.7R

,

-

\h~? + (O.77rnD)3! /'

\

//

Di~ensionaLAtialysis

The 'laws of similarity fo~ propellers may be obtained directly by dimensional analysis, though .without the insight provided by the analysis of the preceding section. If the thrust T of a propeller depends upon its size as characterised by its diameter D, its speed of advance ~ and revolution rate n, the density p and viscosity J.L of the fluid, the acceleration due to gravity g and a suitably defined pressure P, one may write: '

T = f(D,

~,

n, p, J.L, g, p) . (4.5)

The Propeller in_ "Open H Water or, in terms of the dimensions mass M, length

L. and time 'I.:

t

[111 1:: T- 2 ] = [L.l i1 [1:: T- 1 ]i [L- 1t [M L. -3]i 4 [M L. -1 L- 1 Z

3

s

[~T-2]i6 [M L.- 1T- 2]ii

so that:

and: i1 i2 i4

= 2 + i3 = 2 - i3 = 1 - is -

is

+ i6 -

2i 7

is - 2i6 - 2i 7 i7

Therefore:

or:

(4.6) Multiplying both sides by ]'2 does not alter the nature of the functional relationship, so that one may write:

T

1IA 2

pD 2 VA 2 n 2 D2

.J..

_

T = --------, pn 2 D4

(4.7)


66

One may similarly show that: Q

- f ( J, Rn, Fn , En)

(4.8)

All the parameters on the right hand side in Eqns. (4.7) and (4.8) are dimensionless numbers l and the thrust and torque coefficients of a propeller are thus defined as: (4.9)

The thrust and torque coefficients along with the advance coefficient J and the 'open 'Yater' efficiency: (4.10)

4.4

Laws of Similarity in Practice

Consider a full size ship propeller of diameter D s operating at a speed of advance VAS and revolution rate ns, the pressure at the centre of the propeller being ps. The question that arises is at what speed of advance, revolution rate and pressure should a model propeller of diameter D M operate so that the laws of similarity are satisfied. The model scale is given by:

>. -

(4.11)

.j 'j


67

The laws of similarity require that the advance coefficient J, the Reynolds number R.n, the Froude number Fn and the Euler number En for the ship propeller and the model propeller be the same. If the Reynolds number of the model propeller is to be equal to the Reynolds number of the ship propeller: (4.12)

Neglecting any differences between the fluids in which the ship and the model propellers operate, this leads to the requirement: VAM

=

Ds

VAS DM

::=

VASA

(4.13)

i.e. the smaller the model propeller the greater must be its speed of advance compared to the ship propeller. The equality of the advance coefficients requires: (4.14)

that is, nM

::=

VAM Ds ns--- VAS DM

and this in association with the equality of Reynolds numbers, i.e. Eqn. (4.13), leads to the result: (4.15)

. Now, suppose that with the advance coefficients and Reynolds numbers of the model propeller and the ship propeller being equal, the thrust coefficients are also equal, that is:

J(TM =

J..__ I

J(TS

(4.16)


68 or, Ts psn~D~

so that

neglecting any difference between PM and PS. This in association with Eqn. (4.15), gives:

= TSA 4 A- 4

TM

= Ts

(4.17)

The practical implications of Eqns. (4.13), (4.15) and (4.17) become clear if one considers an example. Let a ship propeller of 5 m diameter have a thrust of 500 kN at a speed of advance of 10 m per sec and an rpm of 100, and let the model propeller have a diameter of 20 em. Then: A -

\.}(./

5m =20cm

Ds .DM

= 25

VAM - VASA

= 10 ms- 1 X 25

nM - nsA 2

= 100 rpm X 25 2

TM

=

Ts

=

= 250 ms

1

= 6?500rpm

(4.18)

500kN

An experimental facility capable of achieving these values is practically impossible. On the other hand, if the Proude number of the model propeller is to be equal to the Froude number of the ship propeller, VAM VAS

JgDM = JgDs

so that: (4.19)

69


Comparing this with Eqn. (4.13), it is clear that the condition for equal Reynolds numbers and the condition for equal Froude numbers cannot be satisfied simultaneously unless A = 1, i.e..the model propeller is of the same size as the ship propeller. If with equal Froude numbers the advance coefficients of the model propeller and the ship propeller are also made equal, then: VAM D s -05 05 nM = nS-- - - = nsA . A = nSA . , (4.20) VAS DM

If ".rith equal Froude numbers and advance coefficients, the thrust coe1lrcient of the model-propeller is ~qual to the thrust coeffi~ient of the ship propeller: TM

=

Ts

(:~)2 (~~)4

= TSAA-

4

= TSA- 3

(4.21) .

This means that a 20 cm diameter model oLa ship propeller of 5 m diameter producing 500 kN thrust at 10 m per sec anq 100 rpm must have the following values:

A VAM

=

Ds DM

-

VAS >.-0.5

= 25 =

~ =

lOms- 1 x

2.0m~-1

\

\\,\ l

..'

,/ / ~v

\\

nM

= ns AO. 5 =

100 rpm x 5

TM

= TSA- 3 =

500kN

'\ 't

X

=

1 25 3

500 rpm

l'

(4.22)

= 0.032kN

These values can be achieved in practice with comparative ease. Since Reynolds similarity and Froude similarity cannot simultaneously be achieved and Reynolds similarity is almost impossible, it is usual in model experiments with propellers to satisfy only Froude similarity and to make such corrections as are necessary to account for the difference between the Reynolds numbers of the ship propeller and the model propeller. A similar situation exists for model experiments regarding ship resistance. The laws of similarity also require that the Euler number of the model propeller be equal to the Euler number of the ship propeller:

• t

Bask Ship Propulsion

70 POM V2 '2~M AM 1

-

Pas V2 '2 PS AS 1

or POM

:::::

Pos

(VAM '\2 = Pas \ VAS)

>.-1.

(4.23)

If POM and Pos are the hydrostatic pressures, then this condition is automatically satisfied because of the geometrical similarity between the model propeller and the ship propeller, since the hydrostatic pressure is proportional to the depth of immersion and hence to the propeller diam eter. It is permissible to take the characteristic pressure used in defining the Euler number as the hydrostatic pressure provided that "cavitation" does not occur. Cavitation is discussed in Chapter 6. If there is a possibil ity of cavitation occurring in the ship propeller, it is necessary to take the total pressure minus the vapour pressure as the charaCteristic pressure in the Euler number, which is then called the cavitation number· a. For the ship propeller: PA

+ psghs -

PV

(4.24) "2 PS VAS where PA is the atmospheric pressure, hs the depth of immersion of the ship propeller and PV the vapour pressure. The total pressure for the model pro~eller should then be:

a=

1·

2

(4.25) neglecting the small difference between PM and PS. Special measures are necessary to achieve the value of PaM required by Eqn. (4.25). ,//Example 1 ........

(.~_

A ship propeller of 5.76m diameter, 0.8 pitch ratio, 0.55 blade area ratio, 0.05 blade thickness fraction and 0.18 boss diameter ratio produces a thrust of 1200 kN with a delivered power of 15000 kW at 150 rpm and 7.5 m per sec speed of advance in sea water. The depth of immersion of the propeller is 6.0 m. A 0.16 m diameter model of this propeller is to be tested in fresh water. Determine for the model propeller (a) pitch, (b) blade area, (c) blade thickness at shaft axis, (d) boss diameter, (e) speed of advance, (f) revolution rate, (g) thrust, (h) delivered power and (i) total

Tbe Propeller in "Open" Water

71

pressure if the Froude numbers of the model and the ship propellers are to be made equal. What is the .ratio of the Reynolds number of the ship propeller to the Reynolds number of the model propeller?

Ds

5.76m

P

AE = 0.8 A 0.

D

Ts

=

1200 kN PDs

hs

=

6.0m PS

=

=

------. >.

_~

=

Ds DM

PM

=

d

D = 0.05

15000 kW VAS

1025kgm- 3

PV = 1.704kNm- 2

.'

to

= 0.55

=

5.76 0.16

=

D = 0.18

=

7.5 m s-l ns

150 rpm

=

1000kgm- 3 PA = 101.325kNm'-2

=

36/ ;

From geometrical similarity, one obtains for the model: Pitch, P PM = D

X

DM = 0.8 x 0.16 = 0.128m

Expanded blade area, AEM

AE

= -

Ao

7l'

X -

4

2

7l'

D M = 0.55 x -4 x 0.16

2

2 = 0.01l06m

Blade thickness at shaft axis, tOM

=

(;)DM

=

=

(;) DM

= 0.180 x 0.16 = 0.0288m

0.050 x 0.16

0.008m

Boss diameter,

dM

For equal Froude numbers: .VAM

.Jil5M =

.J..

_

VAS

VgD s '

VAM "~""'---.

2.5 s-l

= VAS >.-0:,5 ..........._ _ _;W>&.'"

1

1 = 75 . x6 = 1.25ms-


72 For equal advance coefficients: VAS

=--, nsDs

= ns).O.5 = 2.5x6

nM

= 15s- 1 = 900rpm

For equal thrust coefficients:

= 1200

=

1000 1025

X

X

(3

1 6)3 leN

0.02509kN

For equal torque coefficients,and noting that PD = 2n n Q,

That is,

PD" - PDS m -

15000 >< 1000 1025

P.'.!).1.5 ).-5

PS

X

36- 3.5 kW = 0.05228kW

., ..",.. ".;':'~,:-c"w~.. ,~ ... ~._,_~

For equal cavitation numbers, PA

POM - PV 1

iPM

+ PS ghs -

PV

V2 iPs AS

V2

1

AAf

that is, PQM

= (p .... +psghs-Pv) .:<~

(101.325

=

__

"(...",~-v;~",",~>l'-....:a..q,;:"

+ 1.025

X

PM ",-l+ pV

PS

9.81

X

1.000 6.0 - 1.704) 1.025

1

X

36

+ 1. 704 kN m

2

t

6.0388 kl\ m- 2

I I

I

The ratio of the Reynolds numbers is:

= ~

1.139 X 10- 6 6 x 36 x ----__::_6 1.188 X 10-

J

=

207.091 .

V? Because of this larg-e difference between the Reynolds number of the ship propeller and the Reynolds number of the model propeller, there will be

.j

I I

1 I

~


73

differences between the thrust and torque coefficients of the ship and model propellers. When considering the performance characteristics of a propeller in open water, some simplifications are usually made in Eqns. (4.7) and (4.8). It is known that the Froude number governs the gravity waves generated at the free surface due to the motion of a body in" a fluid. If the body is submerged sufficiently deep in the fluid no waves are generated at the free surface and the Froude number ceases to influence the flow. It has been observed that if the immersion of the propeller centre line below the surface of water is at least equal to the propeller diameter, the Froude number can be omitted from Eqns. (4.7) and (4.8) without significant error. Further, since the model and ship propeller Reynolds numbers cannot pe made equal in any case, the Reynold,s number is also omitted from the~e equations, and a correction made for this seperately.

If the phenomenon of cavitation is present, the Euler number must be put in the form of the cavitation number, Eqn. (4.24). Based on these consider ations one may write: I

{~~} =

f(J,cr)

(4.26)

Frequently, the possibility of cavitation can be eliminated, and then:

{~~} =

\

f(J)

(4.27)

the exact nature of the function depending upon the geometry of the pro peller. Eqns. (4.26) and (4.27) being relations between dimensionless quantities should be independent of the size of the propeller. Unfortunately, this is not strictly correct because the Reynolds number, which has been neglected in these equations, depends upon the size of the propeller. The correction that must be made for the difference between the Reynolds number of the ship propeller and the Reynolds number of the model propeller is usually small provided that the flow around the model propeller is turbulent in nature just as the flow around the ship propeller. This requirement is met by not making the model propeller tao small, giving it a dull matt surface

'b

.' 74


finish and ensuring that the Reynolds nwnber is above a certain critical value. In order to further reduce the Reynolds number correction, the model propeller in open water is run at as high an axial speed and revolution rate as possible for the required range of advance coefficient, so that the difference between the Reynolds numbers of the model propeller and the ship propeller is minimised. A method to determine the Reynolds number correction for propeller open water characteristics is given in Chapter 8.

4.5

Open Water Characteristics

a

The open water characteristics of propeller are usually given in term~ of the advance coefficient J, the thru,st coefficient KT, the torque coefficient KQ and the open water efficiency 110. These values are given in the form of a table, or KT, KQ and 1}0 are plotted as functions of J. A typical KT-KQ diagram is shown in Fig. 4.1. The values of KQ are usually a little more than one-tenth the values of KT at the same values of J, so that it is convenient

0.8

r---------------.----------, SERIES .... Z =.4 P/O = 0.8 AE lAo = 0.5

0.7 0.6 0.5 0.4

11'o 10K 0.3

o

K

r

0.2

0.1

1

o

J, 00.1

0.2

0.30.4

0.5

0.60.7

0.8

J.

0.9

1.0

1.1.1.2

f

f

Figure 4.1: Kr-KQ Diagram.

i


75

to plot 10 K Q rather than KQ in the diagram. The KT-KQ diagram has some interesting fe.atures. KT and KQ have their largest values at J = 0, i.e. when the propeller is revoh'ing about its own axis without advancing through the water (VA = 0). This condition of propeller operation occurs, for example, when a tug just begins to tow a stationary ship or during the dock trial of a new ship, and is known as the static condition. It is also known as the bollard pull condition from the trial in ~hidl a tug is attached to a ballard by a tow rope and the ma.ximum pull of which the tug is capable is determined. This static condition also corresponds to the 100 percent slip condition discussed in Sec. 2.4. In this condition, with If;t. = 0, a typical propeller blade section has the highest angle of attack equaitothe pitch angle as §Qown in Fig.4.2(a), and this results in KT and KQ having their largest values. (Only values of J greater than or equal to zero are considered here).

The value of J at which KT = a is also of interest. In this condition, the resultant velocity VR may be regarded as being directed along the no-lift line of the representative blade section, Fig. 4';'2 (b) , and since no lift is developed there is no thrust. (Strictly, this is true only if the pegative contribution of drag to the thrust is neglected). This condition of propeller operation is

2nnr

Figure 4.2: Blade Section Velocity Diagram$ at 100 percent Slip and Zero Slip.

-


76

known as the feathering condition. It also corresponds to the condition of zero slip, Sec. 2.4, and since the effective slip ratio is given by:

P

Be

by putting

..

Be

_

=

VA

Pe -J D nD = D Pe Pe D D

-e - - -

(4.28)

= 0 it is se~_~:'::~'::~~~2t:!~,~!-~r~_12~~,,!}.!!.ffi~iS2'llx

e~_~:=:U.9 ~h~,~jf~j:g¥.e_'p'j,tJ:~,~~J~q.~·

A propeller normally operates between the zero and the 100 percent slip

conditions. In botil these conditions the open water efficiency '1]0 is zero.

The maximum value of 1]0 occurs at an effective slip ratio of between 10

and 20 percent, and for values of J greater than that corresponding to the

maximum '1]0, the value of 1]0 falls sharply to zero.

~kample

2

The open water characteristics of a propeller of 0.8 pitch ratio are as follows:

\

o

0.2000

0.4000

0.6000

0.8000

0.3400

0.2870

0.2182

0.1336

0.0332

0.4000

0.3568

0.2905

0.2010

0.0883

, If these results are obtained by running a model propeller of 0.2 m diameter at 3000 rpm Over a range of speeds in fresh water, determine (a) the power of the motor required to drive t he propeller (neglecting losses), (b) the maximum thrust, (c) the maximum open water efficiency, (d) the speed at which maximum efficiency occurs, (e) the speed at which the propeller has zero thrust, and (f) the effective pitch factor, i.e. the ratio .of the effective pitch to the face pitch of the propeller.

DM = 0.200m

nM

= 3000rpm = 50s- 1

p

= 1000kgm- 3

The maximtlIE.p.?~e~ and thrust ()ccur at...J = 0, for which J(r = 0.3400 and = O'"JJ4000. The power required and the maximum thrust are then:

J(Q

_


77

"

= lO.0531kW

= 1.3600kN By calculating

7]0

= KKT

Q

!-. for different values of J, one obtains: 211' I

= 0.6153 at J = O.6?50. Corresponding speed VA = JnD = 0.6650 x ~O x 0.2ms-

Ma:\:imum 7]0

1

= 6.650ms- 1

:I

K T = 0 at J VA

=

= 0.860,

.

and the corresponding speed· i

JnD = 0.860 x 50 x 0.2

Effective pitch ratio Face pitch ratio Effective pitch factor

Pe D

P

D

=

8.6ms- 1

= 0.860

= 0.800 0.860 = 1.075 0.800

A propeller is normally used to propel the ship ahead, Le. the speed of advance VA and the revolution rate n of the propeller are positive. However, the propeller may be run in the reverse direction to propel the ship astern (VA and n both negative). 'The propeller may also be reversed to decelerate the ship when it is going forward (VA positive, n negative). When a ship going astern is to be stopped the propeller is run in the forward direction, Le. ~ negative and n positive. The open water characteristics of a propeller for both directions of advance and revolution, ",foUl' quadrant characteristics", are illustrated in Fig. 4.3.


78

3.0

.,

\ \ \

2.0 ......

+

1.0

t 10 K Q Kr

i

~ 0

"" "', ,

10 K Q .

K" r

+

t

I

-n -

-1.0 v

~ '

Kr

- / I

/

I I I

-2.0

i/ /

\.\.

""' foor

/:~ V/ 1/

/

......

L - .....

~ //

-

-- 1.-' /

-~

~ '\~ /

,,

\'"

I 10 K Q

I

/ / \

- _ VA

,-3.0 -2.0

~1.0

-+

o

1.0

2.0

J Figure 4.3: Four Quadrant Open Water Characteristics.

4.6

Methodical Propeller Series

In a systematic or methodical series of propellers, all the propellers belonging to the series are'related to one another according to a defined "system" or "method". Then, by determining the open water characteristics of a sm.all number of propellers of the series, the characteristics of any propeller of the


79

series may be easily calculated. Generally, only the gross parameters of the propeller such as pitch ratio and blade area ratio are systematically varied. Details such as blade section shapes (camber ratio and thickness distribution) are kept unchanged. Methodical propeller series data are widely used in propeller design. Several methodical propeller series have been developed over the years. Two such series that have been widely used in propeller design are described in the following. The Gawn or the AEW (Admiralty Experimental Works) 20-inch method ical series (Gawn, 1953) consists of propellers that have elliptical developed blade outlines and segmental blade sections. The parameters that are sys tematically varied in the Gawn series are the pitch ratio and the developed blade area ratio. The major particulars of the Gawn series propellers are given in Table4.1 and Fig. 4.4. The open water characteristics of the pro pellers are given in the form of KT-KQ diagrams. Each diagram contains KT, K Q and 110 curves for P/D = 0.6,0.8,1,0, ... 2.0, and there are different diagrams for AD/Ao = 0.20; 0.35, 0.50, ... ,1,10. Expressions for KT and KQ as polynomial functions of J, P/ D and AD/Ao are given in Appendix 3. ;

Table4.1· Particulars of Gawn Series Propellers

No. of blades Pitch ratio Blade area ratio (developed) Blade thickness fraction Boss diameter ratio

Z = 3

P/D = 0.60-2.00 AD/Ao = 0.20-1.10 tolD

0.06

diD

= 0.20

Another noteworthy methodical series of propellers is the B-series of MARIN, also known as the Troost, Wageningen or NSMB B-series, Oost erveld and Oossanen (1975). The B-series has been developed over several years, beginning with the results presented by Troost (1938). Propellers of the B-series are described by a number indicating the number of blades fol lowed by one indicating the expanded blade area ratio, e.g. a B 4.40 propeller

J..

_

80

Basic Ship Propulsion A

b FOR

o A:C o

!"

0.35

~O.010

0.03750

0.03750

0.107~~ ~=;:_·-l~-+·--110.;8750 0.060

~---t~-i

= 1.10

0.250

Figure 4.4: Gawn Propeller Series Geometry.

PER CENT

-~~=~~S~~====l100.0 00.0 100.0

-~---r---0.95R1.0R - - " \ \ - - - - ' - - - 0.9R

\

- - - - 4 r \ - - - ! - - - 0.8R

100.0

- - - - 0.7R ---+'\--~-- O.6R - .

100.0 100.0

----\-+-+--- 0.5R

99.2

0.4R

95,0 - , ' - - t

- - - - + - - \ 1 - - 0.3R

88.7 "T------lr

-;::===;;~~=~0 • 2R

82.2 +---, -80.0

'7l

._.. _._.J._._._.

d

\

PITCH VARIATION FOR FOUR BLADED PROPELLERS

~o Figure 4.5: B-Series Propeller Geomet1",

is a B-series propeller with Z = 4 and AE/Ao = 0.40. Fig. 4.5 shows some of the features of the B-series propellers: an asymmetric wide-tipped blade

J


81

outline, and aerofoil sections at the inner radii changing gradually to segmen tal sections at the blade tip. In the B-series, the parameters that have been varied include the number of blades, the expanded blade area ratio and the pitch ratio. The range of the variation of the blade area ratio depends upon the number of blades, and is given in Table 4.2 along with the other main particulars. All the propellers have a constant pitch, except for the four bladed propellers, which have the pitch reduced by 20 percent at the blade root. Further geometrical details of the B-series ,are given in Appendix 3. The open water characteristics of the B-series propellers are available in a variety of forms including diagrams giving KT and KQ as functions of J for PID = 0.5,0.6,0.8,1.0,1.2 and 1.4, with different diagrams for the different values of Z and AE/Ao. The values of KT and KQ have al,so been put into the for;m ofpolynomia1s:

.~ GT(i,i,k,qr (~)j (~~)k Zl

KT =

',J,k,1

KQ =

I

,

(4.29)

.~ GQ(i,j,k,l)r (~)j (1~)k Zl.

',J,k,1

The values of GT and Gq are also given in Appendix3.

Table 4.2 Particulars of B-Series Propellers

L

Z

PID

AEIAo

tolD

diD

2

0.5-1.4

0.30

0.055

0.180

3

0.5-1.4

0.35-0.80

0.050

0.180

4

0.5-1.4

0.40-1.00

0.045

0.167

5

0.5-1.4

0.45-1.05

0.040

0.167

6

0.5-1.4

0.45-1.05

0,.035

0.167

7

0.5-1.4

0.55-0.85

0.030

0.167

82


4.7

Alternative Forms of Propeller Coefficients

Although the KT-KQ-J coefficients are the normal mode of presenting the open water characteristics of propellers, other coefficients have been devel oped which are rr:ore convenient to use, especially for propeller design and performance analysis using methodical series data.. The major difficulty of using the KT-KcrJ coefficients is that all of them contain both the pro peller revolution rate n and the diameter D, and in propeller design at least one of these varic.bles is initially unknown and is to be determined during the design proces~. This makes propeller design using the KT-KQ diagrams of a methodical propeller series a process of trial and error to determine the optimum design parameters. Table 4.3 gives the values of KT and KQ for a hypothetical methodical series in an abbreviated form. Example 3 A propeller running at 126 rpm is required to produce a thrust of 800 kN at a speed of advance of 12.61 knots. Determine the optimum diameter and pitch ratio of the propeller and the delivered power in open water. The propeller belongs to the methodical series for which the open water characteristics are given in Table 4.3.

n == 126rpm

=

2.1s- 1

T = 800kN

VA = 12.61 k = G.48GG m s-1

\

The :straightforward way to solve this problem is to assume different propeller diameters D, calculate the corresponding values of KT and J, and determine by interp~lation betwe.:-n the KT-J and KQ-J curves for the different pitch ratios, the values of P/D, Kc; and hence 7]0 for each assumed value of D. Plotting 7]0 and PI D as functions of D then enables the optimum diameter and the corresponding pitch ratio to be determined. A slightly different approach, which avoids the naed for interpolation between the KT-J and KQ-J curves, is used here. 2.1 2 X 800 = 1.9442 1.025 X G.4866 4

KT

J4

The values of K T and J corresponding to this value of K T /J4 are: J

0.40

0.45

0.50

0.55

O.GO

0.65

0.70

f{r

0.0~97

0.0797

0.1215

0.1779

0.2520

0.3470

0.46G8

-

--------,

Table 4.3

b2 (1)

~

Open Water Characteristics of a Methodical Series

Z=4 0.5

P/D: J 0 0.1 0.2 0.3 ,0.4 0.5 0.6 0.7 0.8 0.9 1.0

KT

0.2044 '0.1826 0.1795 0.1642 0.1499 0.1423 0.1156 0.1168 .0.0765 0.0879 0.0327 0.0553

-

-

-

-

-

10KQ

0.2517 0.2254 0.1949 0.1603 0.1215 0.0786 0.0315

0.2455 0.2247 0.2001 0.1718 0.1369 0.1036 0.0639

-

-

-

-

-

=:::

0.8

0.7

KT

-

Cb

AE/Ao = 0.500 -

0.6

10KQ

.g

KT

10KQ

0.2974 0.3187 0.2702 0.2956 0.2393 0.2684 0,2047 0.2373 0.1665 0.2021 0.1246 0.1629 0.0790 0.1197 0.0297.0.0725

-

-

-

-

....::J~

0.9

KT

10KQ

KT

10KQ

0.3415 0.3114 0.2831 0.2489 0.2115 0.1707 01266 0.0792

0.4021 0.3767 0.3471 0.3i33 0.2753 0.2331 0.1867 0.1361 0.0813 -

0.3840 0.3567 0.3263 0.2929 0.2564 0.2169 0.1743 0.1287 0.0800 0.0283

0.4956 0.4681 0.4362 0.3999 0.3593 0.3143 0.2649 0.2112 0.1530 0.0906

0,02~'4

-

~§ ~ .... (1)

""I

00

C>:l

00

Table 4.3 (Contd.)

PID

/1.1

1.0

1.2

.J

10[(Q_

[(T

_"!!!!~9

. lfT..

o

0.5994 0.5698 0.5357 0.4971 0.4540 0.4064 0.3543 0.2976 0.2365 0.1708 0.1007

0.4644 0.4391 0.4109 0.3799 0.3461 0.3095 0.2701 0.2278 0.1827 0.1348 0.0841 0.0305

0.7133 0.6818 0.6456 0.6048 0.5594 0.5094 0.4548 0.3955 0.3317 0.2632 0.1900 0.1123

0.5022 0.8374 0.4787 0.8040 0.452.3 0.7659. 0.4231 0.7231 0.3910 0.6757 0.3560 0.6234 0.3181 0.5665 0.2774 0.5049 0.2338 0.4385 0.1874 0.3675 0.1381 0.2917 0.0859 0.2113 0.0308 0.1261

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

~

} g!SSJ

1.3

._!
_

1.4 10[(Q

[(r

19[(Q

0.5384 0.9717 0.5730 1.1161 0.5173 0.9366 0.5548 1.0794 0.4931 0.8967 0.5333 1.0378 0.4959 0.8520 0.5086 0.9915 0.4358 0.8026 0.4805 0.9403 0.4026 0.7484 0.4492 0.8843 0.3663 0.6894 0.4146 0.8235 0.3271 0.6257 0.3768 0.7579 0.2848 0.5571 0.3357 0.6875 0.2396 0.4839 0.2913 0.6122 0.1913 0.4058 0.2437 0.5322 0.1400 0.3229 0.1928 0.4474 0.0857 0.2353 0.1386 0.3577 0.0284 0.1429 0.0812 0.2632 0.0205 0.1640

b:l

!Jl

n' ~ "6'

1 l:

r;;

g'

~

• . • as

'~':'~

Q

•

~~.-:o

._

-

~

,

,,,"'8""=###


85

Plotting this on the KT-KQ-J diagram obtained from the data in Table 4.3, one obtains at the points of intersection with the KT-J lines for the different pitch ratios the values of J and KT given in the following table. The corresponding value of K Q is obtained from the KQ-J lines, and the value of 1]0 calculated. P D J

0.6

0.8

1.0

1:2

0.5327

KT 10KQ

0.4676 0.0930 0.1156

0.1566 0.2184

0.5854 0.6299 0.2283 0.3063 . 0.5486 0.3622 ,

'70

0.5987

0.6079

0.5872

0.5597

! Finally, .plotting '70 as a function of P j D, tqe maximum' efficiency is found to be 0.608, corresponding to the optimum values: J = 0.521, lOKQ = 0.196, PjD = 0.762. Then, the optiIDum diameter and the corresponding delivered power are obtained: . p 6.4866 = 0.762 = 5.9~9m 2.1 x 0.521 D I

= 8565kW Several other forms of propeller coefficients have been developed of which two sets of coefficients that have been. widely used are considered here. The Bp - 6 coefficients were introduced by Admiral D. W. Taylor and are defined as follows: Bp

6

=

(4.30)

where:

D

-

propeller diameter in feet;

n

=

propeller rpmj

PD

-

delivered power in British horsepower (1 hp = 0.7457 k\iV) for the open water condition in fresh water!

"A

=

speed of advance in knots.

1_


86

, i

A typical Bp-8 chart, as shown in Fig. 4.6, consists of contours of 8 and on a grid of B p' and P/ D. The values of B p are usually plotted on a square root scale, i.e. the horizontal scale of the Bp-8 diagram is linear in VB p. The "optimum efficiency" line shown in the B p-8 diagram indicates the point at which 7]0 is maximum for a given Bp. The Bp-8 diagram is convenient to use when the speed of advance ~, the propeller rpin nand the delivered power PD are known and the propeller diameter D and pitch ratio P/ D for optimum efficiency are to be detern:tined. If, however, instead of the delivered power it is the thrust power PT = T VA that is known, i't is more convenient to use the .coefficient Bu defined as: 7]0

B

-

U -

n p,0.5

(

T

(4.31)

V2.5 A

where PT is the thrust power in hp for the open water condition in fresh water. The system of propeller coefficients introduced by Admiral Taylor includes a number of other coefficients, but methodical series data are usually available only in the form of Bp'-8 diagrams.

I

! I

Since the Bp-8 coefficients involve the use of specific units (feet, rpm, hp and knots), the values of parameters expressed in different unit~ have to be first converted into these units, and since the coefficients are given for fresh water, parameters such as delivered power have also to be corrected to fresh water. These difficulties are avoided if, instead of the Bp-8 coefficients, truly dimensionless parameters are used. It can be shown that; Bp = 33.053

(~~) 0.5

and

8 =

10~26

(4.32)

and, therefore, the dimensionless equivalent of the B p-8 diagram consists of contours of 7]0 and 1/ J on a grid of (KQ / J5 )0.25 and P / D. Example 4 A propeller absorbs a delivered power of 7500 kW in open water at 120 rpm the speed of advance being 12.0 knots. Using the Bp-o diagram in Fig. 4.6, determine the optimum diameter of the propeller, the corresponding pitch ratio and the pro peller thrust..

PD = 7500kW

n = 120 rpm

l-A

= 12.0k = 6.1728ms- 1

~ (l)

::p

.g

~

~

S·

~ ~

~ ~

p

o Bp

5

10

20

./

Figure 4.6: Bp- 0 Diagram.

30

40

50

60

00

--1


88

For using the Bp-o diagram, the delivered power must be converted to fresh water, and British units used.

==

PD

7500kW

1

o.7457 W up n po. 5

v~r:s ==

Bp ==

X --

1.025 120

== 9812.35 hp (Fresh water)

9812.350.5 12.02.5

X

== 23.83

From the optimum efficiency line in the Bp-o diagram (Fig. 4.6), one obtains: p

D

o=

== 0.798

'fJo = 0.593

190.5

Hence: D

= ~6 = n

12.0 ~x 190.5 = 19.05 ft = 5.806 m 120

T = PD'fJo == 7500

0.593 = 720.5kN 6.1728

~

X

For propellers which operate at very high slip ratios (small values of J or high values of 0), such as tug and trawler propellers, the Bp-o diagram often cannQt be used since the values of Bp and 0 fall outside the normal range. In such cases, one may use the /.L-cr system of coefficients due to F. Gutsche (1934):;

pn2 D5)0.5 I/.

-

,... - (

Q D3)0.5

CP =. VA ( PQ

cr

cr.

_

- K Q 0.5

-

== J K'Q°·5

(4.33)

DT == 21rQ

A /L-cr diagram consists of contours of T]O, cp and P/ D on a grid of It and A typical/L-cr diagram is shown in Fig. 4.7.


89

0.8

O. 7 O. 6 O. 5 O. 4 O. 3

0 o. 2

2

4

5

6

7

8

9

Figure 4.7: f.1-C5 Diagram.

10

11

12

13


90

The use of computers for propeller design and analysis has greatly reduced the need for special propeller coefficients aimed at reducing the amount of calculation involved. Computer programs for propeller design usually make direct use of the KT-KQ-J coefficients. Example 5

A propeller of 3.0m diameter and 0.8 pitch ratio runs at 150rpm at zero speed of advance. If the engine driving the propeller produces a constant torque, find the propeller rpm, delivered power and thrust at the following speeds of advance: 0,3, 6, 9 and 12 knots. Use the J.J.-(1 diagram of Fig. 4.7.

D= 3.0m

j.L=

(pn~D5) 0.5 =

If' = ~

u=

AX ~ =

P

D == 0.8

(Pg

3 ) 0.5

=

At

(1.025

~

n

X

~2 x 3.05) 0.5

(1.025; 3.0

3 )

0.5

= 150 rpm

= 2.5 s-l

= 15.7821nQ-O.5

= 5.2607 VA Q-0.5

3.0 x T DT = 0.4775 T Q-1 = 211" X Q 211"Q

0, n = 2.5 s-l, and If' = O. For If' = 0,

~

= 0.8, from the j.L-u diagram:

j.L = 4.945 so that:

15.7821 x 2.5 x Q-0.5 = 4.945

and:

Q = 63.6615 kN m

Hence:

j.L = 1.9780n
= 0.6593 VA

u = 0.0075 T

n == 0.5056j.L PD = 211"nQ

T == 133.33 u

, The Propeller in "Open" Water

91

'r

IA k ms- 1

0 0

.p

'3

6

9

12

1.5432

3.0864

4.6296

6.1728

O.

1.0174

2.0349

3.0523

4.0697

/1

4.943

5'.264

5,713

6.292

6.929

p

,1.329

1.270

1.206

1.136

1.057

n s-1

2.499

2.661

2.888

3.184

3.503

rpm

150

160

173

191

210

;1000

··'1064

1155

1274

1401

179.3

'160.8

151.5

140.9

?DkW

TkN

177.2

Problems, 1.

A ship propeller of 5.0 m diameter has ,a thrust of 500 kN and a torque of

It 375 kN m at a 'revolution rate of'120 rBm and a speed of advance of 6.0 m per sec. Determine the speed of advan<;:e, 'rpm, thrust and torque of a model propeller of diameter 0.25 m if the Protide numbers of the ship propeller and the model propeller are equal. If the depths of ipunersion of the ship and model propellers are 6.0m and 0.3 m respectively, what is the ratio of their cavitation numbers? What is. the ratio of the Reynolds number of the ship propeller to that 'of the model propeller? 2/ A ship propeller of 4.0 ~ diameter h~ a thrust of 160 leN and a torque of fj 120kNm when running at,150rpm and 8.0m per sec speed of advance in open water. In order to make the Reynolds number and the Proude number of the ship propeller respectively equal to the Reynolds number and the Proude number of the model propeller; it is proposed to test the model propeller in a liquid ~hose density is 800 kg per m 3 and kinematic viscosity 1.328 x 10- 8 m 2 per sec. Determine the diameter of the model propeller and its speed of advance, rpm, thrust and t~rque. (Note, Unfortunately, no iiquid anything like this exists.) 3.

A model propeller of 0.15 ill diameter is run at 3600 rpm over a range of speeds of advance and the following values of thrust and torque are recorded:

Speed of Advance m pel' sec

a 1.000

-

Thrust N

Torque Nm

770.71 715.94

16.930 15.850

92

Basic Ship Propulsion Speed. of Advance m per sec 2.000 3.000 4.000 5.000 6.000 7.000 8.000 9.000

Thrust N

Torque Nm

654.83 587.55 513.93 433.95 347.64 254.97 155.96 50.78

14.654 13.338 11.904 10.352 8.684 6.897 4.993 2.970

Obtain the open water diagram for the ship propeller given that the values of J(T for the ship propeller are 0.5 percent higher and the values of KQ 1.5 percent lower than the corresponding values of the model propeller because of the differences between the ship and model propeller Reynolds numbers. Show that the ship propeller has a maximum. efficiency of 0.7016 at an advance coefficient of 0.797. What is the effective pitch ratio of the ship propeller? 4. A propeller is required to absorb a delivered power of 10000kW at 150rpm when advancing into open water at a speed of 10.0 m per sec. Determine the I optimum. diameter and pitch ratio of the propeller using the methodical series data given in Table 4.3. What is the thrust of the propeller? 5. A propeller of 4.0 m diameter is required to produce an open water thrust of 300 leN at a speed of advance of 6.0 m per sec. Determine the rpm of the propeller for it to work at the optimum. efficiency and the corresponding pitch ratio and delivered power. Use the data of Table 4.3. 6. 'A propeller of 3.0 m diameter and 0.8 pitch ratio absorbs its maximum. delivered power when running at 180 rpm at a speed of advance of 13.5 knots. If the propeller torque remains constant, determine the propeller rpm, thrust· and delivered power at speeds of advance of 0, 2.5, 5.0, 7.5 and 10.0 knots, using the data of Table 4.3. 7. A propeller is required to produce a thrust of 500 leN when running at 150 rpm and a speed of advance of 15.5 knots in open water. Using the Bp-8 diagram given in Fig. 4.6, determine the optimum diamete~ and pitch ratio of the propeller and the corresponding delivered power. 8. A propeller of diameter 5.0m is to have a delivered power of 7500kW at a speed of advance of 15.0knots in open water. Using the Bp-8 diagram of Fig. 4.6, determine the rpm of the propeller for optimum. efficiency as well as its pitch ratio and thrust. 9. A propeller of 3.0 m diameter absorbs a delivered power of 1400kW at 210rpm and 12.0 knots speed of advance in open water. The propeller torque is

----_._---------- ..

.~ ..."'f


93

constant. Using the /l-q' diagram of Fig.4.7, determine the pitch ratio of the propeller, and calculate the rpm, delivered power and thrust at speeds of ad':ance of 0, 3, 6, 9 and 12 knots. . 10. A propeller with a delivered power of 1000kW at 120rpm is required to produce the maximum thrust at zero speed of advance in open water. Using the /l-(f diagram of Fig. 4.7, determine the diameter /W.cI.'pitch ratio of the propeller (within the limits of the diagram). If the propeller diameter is not to exceed 3.5 m, what will be the pitch ratio and the corresponding thrust?

CHAPTER

5

The Propeller "Behind" the Ship 5.1 Introduction When a propeller is fitted at the stern of a ship, it operates in water that has been disturbed by the ship during its forward motion. The behaviour of the propeller is therefore affected by the ship. In a similar manner, the operation of the propeller "behind" the ship hull affects the behaviour bf the ship. There is thus a mutual interaction between the hull and the propeller, each affecting the other. This "hull-propeller interaction" has three major effects on the performance of the propeller and the ship taken together as compared to their behaviour considered individually: (i) +,he propeller advances into water disturbed by the ship so that the flo~v is not uniform over the propeller disc, and in general the speed of, adv:ance of the propeller VA (averaged over the propeller disc) is different from the speed of. the ship V with respect to undisturbed w}l.ter. The disturbance behind the ship due to its motion ahead is ... called "wake". ~

(ii) The action of the propeller alters the pressure and velocity distributions around the hull, and there is therefore an increase in /,:r the total resistance RT of the ship at a given speed V with the propeller working compared to the resistance when there is no propeller. (The resistance of a ship at a given speed is the force opposing'the steady ahead motion of the ship in calm water.) The 94

\

"

The Propeller "Behind" the Ship

95

increase in the resistance of the ship due to the action of the propeller, or "resistance augment" is' for convenience replaced by an equivalent decrease in the propeller thrust and called "thrust deduction" .

/'

(iii) The efficiency of the propeller in the "behind" condition when working in non-uniform flow is different from its efficiency T/o in open '1 water (uniform flow). The ratio of the propeller efficiency in the behind condition to the open water efficiency is called "relative rotative efficiency".

5.2

Wake

The effect of the disturbance created behind the ship by its forward motion is to impart a velocity to the particles of water in the wake. This wake velocity arises basically from three causes. The first cause may be explained by regarding the ship to be moving in an inviscid fluid so that the flow around the ship can be regarded as potential flow and the streamlines around the hull determined .accordingly, Fig. 5.l( a). The nature of the streamlines indicates that at the bow and the stern there are stagnation points (zero fluid velocity relative to the hull), so that the forward and after ends of the ship are regions of low relative velocity. Q...9n side~illKtlJ..efltlid to be at rest Jar from the ship, a low relative velocityat the stern means a high wake velocity directed forward in addition to the velocity components' normal to .the direction of IIl.otionof the .shi,p:·Th~ disturbance" in Hie flow due to this cause is called potential or streamline wake. If now one considers the ship to be moving in water, then due to the viscosity of water a "boundary ,layer" forms over the hull surf
L

t\

.i" :~

:,


96

_._._~..!--..

--~._.

~~~:~'~=~:"=?~5J~~~

(0) STREAMLINES !N POTENTIAL FLOW

Velocity Profiles

I

Boundary Layer (b) VELOCITY PROFILES IN THE BOUNDARY LAYER

\

==liW===~-3/-~\

.

~

Orbital Velocities

(c) ORBITAL VELOCITIES IN A WAVE Figure 5.1 : Origin of Walle.

The third cause for the existence of wake velocity arises from the waves generated by a ship moving at the surface of water. Waves in deep water cause the particles of water to move in circular orbits, the radius of the orbit decreasing exponentially with depth below the surface. The orbital velocities of the water particles constitute the third component of wake velocity, and

The' Propeller "Behind" the Ship

97

this is called wave wake~ The orbital velocities are directed fonvard below a wave crest and aft below a wave trough, Fig. 5.1(c), so that at the location of the propeller the wave wake velocity in the direction of motion of the ship is positive or negative depending upon whether a wave crest or a wave trough accompanies the stern of the ship. In a high speed twin screw ship such as a destroyer, in which the propellers largely.- lie outside the boundary layer of the ship and there is a large wave trough at the stern, the wave wake is often predominant at the propeller and the overall wake velocity is small and may even be negative (Le. directed aft). The wake velocity at the location of the propeller is not uniformly dis tributed and can ,be resolved into three components - a component along the axis of the propeller, and tangential and radial components in the plane of the pr,opeller disc. The tangential velocity components on the port and starboard sides of the ship's centre plane are equal due to the symmetry of the hull, while the radial components are generally small. Therefore, in considering the average stea
v-'( r ) -_

_ ~

f;1l' vCr, e) de fa2 de

-

211"

1l'

= 11"

= I'

.J.

1 . (R2 _ 2)

(R2 2_

i1 R

r~)

a

rb

i

1i"

V

(

r, e)

~

u:u

(5.1)

is given by:

2

1l'

rb

2

p~peller disc

The average wake velocity over the

v

1a

V

(r, B) r dB dr

R

rb

Vi

(r) r dr

(5.2)

where R is the radius of the propeller and rb the radius of the boss. v is called the volumetric ,mean wake velocity sinGe the volume of water flowing through the propeller disc per unit time is equal to 1r (R 2 - r~)v .

~

.. Basic; Ship Propulsion

98

·.1'·.··

, .,

The ratio of the a"erage axial wake velocity v to the ship speed V is called the "wake fraction". w: W=

V

(5.3)

The speed of advance of the propeller is then given by:

VA - V-v - (l-w)V

(5.4)

This definition of wake fraction is due to D. W. Taylor. Another definition, due to R. E. Froude. in which the wake fraction is taken as the ratio of" the wake velocity to the speed of advance is no longer used. The wake Yelocit)~ at the location of the propeller in a ship can be/ mea sured by instruments in the absence of the propeller. The wake f~actioII-v determined by such measurements is called the "nominal" wake fraction. The propeller when fitted at the stern and producing.thrust alters t.h~~ake velocity, and the wake fractiQn determined from the wake velocity;with the propeller operating is called the "effective" wake fraction. Since the wake ve locity at the propeller location cannot normally be measured by instruments in the wake when the propeller is fitted, the effective wake is determined in an indirect manner by putting the speed of advance in the behind condition equal to that speed of advance in open water at which the propeller has the same thrust (or. alternatively, the same torque) at the same revolution rate. When the thrusts in the behind and open water conditions are made equal, the wake fraction is denoted by WT and is said to be based on "thrust identity". If the torques are made equal, the wake fraction wQ is said to be based on "torque identity" .

/Example 1

,,>:-1

A ship has a propeller which produces a thrust of 500 kN with a torque of 450 kN m when the ship speed is 16.0 knots, and the propeller rpm is 120. The same propeller running at 120 rpm produces a thrust of 500 kN when advancing into open water at a speed of 12.8 k, and a torque of 450 kN m at a speed of 13.0 k. Determine the wake fraction on the basis of (a) thrustidentity, and (b) torque identity. .

The Propeller f'Behind" the Ship (a)

VA = 12.8k

For thrust identity, V = 16.0k WT

(b)

99

16.0 - 12.8 = ---16.0

=

For torque identity, V;' 16.0k =

= 0.2000

VA = 13.0k

16.0 -13.0 ---= 0.1875 16.0

In actual practice, the open water data are obtained with the help of a model propeller which may not be run at the same rod,al and rotational speeds in ,open water as in the behind condition, and in that case the wake fraction is determined by making the thrust coefficients in the open water and behind conditions equal, or by making the torque coefficients equal, i.e. by "KT identity" or "KQ identity". Experiments with models of ships and propellers are considered in Chapter 8. I

5.3

Thrust Deduction·

When a propeller produces thrust it accelerates the water flowing through the propeller disc and reduces the pressure in the flow field ahead of it. The increased velocity of water at the stern of the ship and the reduced pressure cause an increase in the resistance of the ship. If RT is the total resistance the total of the ship at a given speed in the absence of the propeller and resistance at the same speed when the propeller is producing a thrust T, then the increase in resistance due to the action of the propeller is:

\

RT

=

6R

R'r-RT

and if the ship speed is constant:

T =

RT =

RT+6R

Since this increase in resistance is an effect due to the propeller, it is convenient to put 6T = 6R and to write: i

1___


100

T-8T = RT

where 8T is the "thrust deduction". The ratio of the thrust deduction to the thrust is called the thrust deduction fraction t: t =

8T T

so that: (1- t)T = RT

(5.5)

The thrust deduction f1"action is related to the wake fraction, a high wake

fr~ction usuallLb_~1J,g.M~Qdat.e.d..with...a..high_thr.ust:...deduction1r.a'C"tIOn~, The

thr~t d~du~tion fraction .~!~(). c.l~I?~~
5.4

. . .. ..

..._ ....:..... .'"

.

Relative Rotative Efficiency

Since the propeller in the open water condition works in undistu.rbed water whereas the propeller behind the ship works in water that has been disturbed by the ship, the efficiencies of the propeller in the two conditions are not equal. If To and Q0 are the thrust and torque of a propeller at a speed of advance VA and re\'olution rate n in open water, and T and Q are the thrust and torque at the same values of VA and n with the propeller operating behind the ship, the propeller efficiencies in open water and behind the ship are respectiYely: "'70 =

To "A 271" n Qo

1]8

=

TVA 271" n Q

and the relative rotative efficiency, which is the ratio of given by: T Qo 77R = - -

To Q

1]8

to

1]0,

is then (5.6)

As discussed in Sec. 5.2, the speed of advance is determined indirectly through thrust identity (KT identity) or torque identity (KQ identity), so that: \


71R

71R

=

Qo Q

=

T To

101

Thrust Identity (5.7) Torque Identity

The relative rotative efficiency is usually.quite close to I, lying between 1.00 and 1.10 for most single screw ships and between 0.95 and 1.00 for twin screw ships. ,/"

~/£xample 2 A ship has a speed of 20.0 knots when its propeller of 5.0 m diameter has an rpm of

150 and produces a thrust of 500 kN at a delivered power of 5650 kW. The resistance of the ship at 20.0 knots is 390 kN, and the open water characteristics of the propeller are as follows:

J

\ 'i· \ c,

0.500

0.550

0.600

0.650

0.700

0.174

0.154

0.133

0.110

0.085

0.247

0.225

0.198

0.173

0.147

Using (a) thrust identity and (b) torque identity, determine the wake fraction, the thrust deduction fraction, the relative rotative efficiency and the open water efficiency.

v

= 20.0k = 10.288ms- 1

T = 500kN

D = 5.0m

n = 150 rpm

PD = 5650kW

Q

=

PD 27fn

RT = 390kN

=

5650 27f x 2.5

=

500 '.

= 0.1249 1.025 x 2.5 2 x 5.04

=

359.690kNm

(a) Thrust Identity:

[(T =

T pn 2 D4

= 2.5 s-l


102

From the open water characteristics for this value of KT:

= 0.617

J

K Q = 0.0191

=

Qo :::: K Q pn 2 D 5 JnD V

l':"'w::::

I-t

=

X

5.0

5

0.617 x 2;5 x 5.0 = 0.7497 10.288

=

382.373 leN m w

390

RT :::: 0.7800 = 500 T

=

382.373 Qo = 359.690 Q

7]R ::::

=

0.2503

1.0631 1

=K Q 2'11"

=

t = 0.2200

.

0.1249 0 617 --x-2n

KT J

7]0

0.0191 x 1.025 x 2.5 2

= 0.0191

= 0.6421

(b) Torque Identity: 359.690

Q

KQ

= ·pn2 D5 =

---~---::-5 1.025 x 2.5 2 x 5.0

= 0.01797

FroIl) the open water characteristics for this value of KQ:

J To

K T = 0.116

= 0.637 = KTpn 2 D 4 = 0.116 x 1.025 x 2.52 x 5.04 = 464.453kN

0.637 x 2.5 x 5.0 = 0.7740 10.288

1-w

=

JnD V

1-t

=

390

RT = 500 T

7JR

=

T To

7]0

KT J =K Q 21l' =

=

=

=

0.7800

500 = 1.0765 464.453 0.637 0.116 x-0.01797 21l'

=

0.6544

w

=

0.2260

t

=

0,2200


5.5

103

Power Transmission

The power delivered to the propeller is produced by the propulsion plant of the ship and is transmitted to the propeller usually by a mechanical sys tem, or some~imes by an electrical system. The propulsion plant or main engine may be a steam turbine, a gas turbine or a reciprocating internal combustion (diesel) engine. A turbine runs at a very high speed and it is necessary to reduce the speed by using a speed reducing deviCe such as me chanical gearing. High speed and medium speed diesel engines also require speed reducing devices, whereas low speed diesel engines do not. The power produced by the main engine is transmitted to the propeller, after speed reduction if necessary, through a shafting system consisting of one or more shafts s~pported on bearings,. The shaft on which the propeller is mounted is called the tail shaft, propeller shaft or screw shaft and is supported by bearings in a stern tube. There is also a thrust bearing to transmit the propeller thrust to the ship hull. In an electrical propulsion drive, the main engine drives an electric generator and the electrical power is transmitted by cables to an electric motor which drives the propeller through the propeller shaft. \

The power produced by a diesel engine may be determined by measuring the variation of pressure in the engine cylinders by an instrument called an indicator. The power so determined is called the indicated power PI. It is also possible to measure the power of the engine by operating it against a load applied through a brake dynamometer. The power determined in this manner is called the brake power PB. The brake power is slightly less than the indicated power due to the mechanical losses that take place within the engine. The brake power is thus the power output of the engine and is carefully measured at the engine manufacturer's works along with the other operating parameters of the engine. The power produced by a steam turbine or a gas turbine is usually de termined by a torsionmeter fitted to the shafting connecting the turbine to the propeller through the gearbox. The power determined by measuring the torsion of the shaft is called the shaft power Ps. A torsionmeter may also be used to determine the shaft power when the ship has a diesel engine. The shaft power varies with the location of the torsionmeter on the propeller

.kI

_

'"


104

shafting, being slightly higher when the torsionmeter is fitted close to the engine than when it is fitted close t~Uhe propeller. The power' that finally reaches the propeller is the delivered power PD, and this is related to the propeller torque Q:

PD = 27rnQ

(5.8)

n being the propeller revolution rate. The delivered power is somewhat less than the brake power or the shaft power because of the transmission losses that take place between the engine and the propeller, i.e. in the gearing and the bearings in mechanical transmission, and in the generator, cables, motor and bearings in an electrical propul~ion drive. The propeller produces a thrust T and this multiplied by the spee~ of advance loA gives the thrust power P2,: I

,i

(5.9)

The propeller thrust causes the ship to move at a speed V overcoming its resistance RT, and the product of the resistance and the ship speed is the effective power PE: \

(5.10) Il may be noted that Eqns. (5.8), (5.9) and (5.10) being dimensionally homogeneous are correct as they stand for any consistent system of units. Thus, if forces are measured in kN, speeds in m per sec, torque in kN m and rcyolutions are per sec, the powers will be in kW. Fig. 5.2 shows a schematic arrangement of the propulsion system of a ship, and the powers available at different points of the system. The brake power PB or the shaft power Ps, depending upon whether the main engine is a diesel engine or a steam or gas turbine, may be regarded as the input to the propulsion syst~m and the effective power P E as its output. If one wishes to focus only on the hydrodynamics qf the system then the delivered power PD is taken as the input.


105

, t. 1. ENGINE

2. REDUCTION GEAR (Optlonol)

.'

3. THRUST BEARING 4. SHAFT 5. BEARING 6. STERN TUBE

Figure 5.2: Propulsion SJlstem of a Ship.

~;jf~D1Ple 3

.

./A ship moving at a speed of 18.0 knots is propelled by a gas turbine of shaft power 10000 kW at 5400 rpm. The turbine is connected to the propeller through 45: 1 reduction gearing. The losses in the gearing and shafting are 5 percent. The pro peller has a thrust of 900kN, and the wake fraction and thrust deduction fraction are 0.250 and 0.200 respectively. Determine the delivered power, the thrust power and the effective power, as well as the propeller torque. ., '

v =

18.0 k = 9.2592 ms- 1

Gear ratio = 45: 1

T

=

900kN

Propeller revolution rate

,175"

Ps = 10000kW

n eng = 5400 rpm

Transmission losses = 5 percent w

=

0.250

t = 0.200

120rpm = 2.0s- 1


106

= 9500kW (1- 0.250) 9.2592 = 6.9444ms- 1

PD = Ps -losses = 10000 (1 - 0.05)

VA

PT RT PE

Q

5.6

= (1::" w) V = = T Y.4 = 900 x 6.9444 = 6250 kW = (1 - t)T = (1- 0.200) 900 = 720kN = RT V = 720 x 9.2592 = 6667kW =

PD 21rn

9500

= 271"

X 2

= 756 kN m

Propulsive Efficiency and its Components

!

As indicated in the previous section, the brake power PB (for diesel engines)

or the shaft· power Ps (for turbines) may be regarded ~ the input/to the propulsion system, and the effectiye power PE as its output. The efficiency

of the system as a whole or the overall propulsive efficiency is then:·

7]OIIerall

\

PE - PE PE = Ps

1

(5.11) (for turbines)

If one considers only the hydrodynamic phenomena occurring outside the

hull, the input to the propulsion system is the delivered power PD, and the

propulsive efficiency is given by:

PE PD

(5.12)

This is sometimes called the quasi·propulsive coefficient qpc. As indicated earlier, the delivered power PD is slightly less than the brake

power PE or the shaft power Ps due to the losses which take place in, the

transmission of power from the main engine to the propeller. The efficiency

....•.

_._---------

1

1I

(for diesel engine)

TID =

i

(

-

,

. I


107

of power transmission from the engine to the propeller is called the shafting efficiency 1]S: 1]S

-

= -PD PB

PD

orPs·

(5.13)

The losses that take place in the transmission of power are usually ex pressed as a percentage of the brake power or shaft power. In installations in which the engine is directly connected to ~he propeller by shafting, the transmission losses are usually taken as 3 percent when the engine' is amid ships and 2 percent when the engine is aft,: With mechanical reduction gearing or electric propulsion drives, the tratismission losses are higher, 4'to 8 percent. The propeller receives the power PD delivered to it and converts it to the thrust power PT when it operates behind the ship, and hence the propeller efficiency in the behind condition is given by:

(5.14) If the propeller were operating in open water at t1.le sa~e speed ofadvanc'e and revolution rate with a delivered power PD~ and a thrust p'owerPTo, then its efficiency in open water would be: \

1]0

=

PTO PD~

(5.15)

The propeller efficiency in the behind condition may be written as:" (5.16)

where:

1]R

=

PT PD~ PTO' PD

(5.17)

is the relative rotative efficiency discusseq in Sec. 5.4. For. thrust identity PT = PTO, while for torque identity PD = PD~, so that:

Jrr

_


108

TJR

PD~

- PD

=

PT PTO

for thrust identity

(5.18) for torque identity

Eqns. (5.6) and (5.17) are equivalent, and 'so are Eqns. (5.7) and (5.18). ".

.

.

. The thr'ust power PT "input" to the hull enables the ship to obtain the effecti~e power PE required to. propel it at a steady speed 'v. The. ratio of . . . ...•. . . I 'the effeCtive 'power to the thrust power is known as the ~ull efficiency:

·T}H -

I-t 1-w

J

(5.:19)

The hull efficiency is usually slightly more than 1 for single screw/ ships a.nd slightly less than 1 for twin sc'rew ~hips. The fact that the hull efficiency and the relative rotative efficiency can have values more than 1 IJ;lay seem somewhat curious, but is explained by regarding these "efficiencies" merely as ratios and not reai efficiencies. From Eqn. (5.19), one may note that for a high hull efficiency, the wake fraction w should be high and the thrust deduction fraction t should be low. If the propeller were placed at the bow of the ship instead of the stern, it would b,e advancing into almost undisturbed water so that the wake fraction would be mllch lower than if the propeller were at the stern. On the other hand, thesllip would be~dvancing into water disturbed by the propeller, and since the propeller accelerates the water flowing through it and increases its pressure, there v.r()~l!lgbeagreater increase in the resistance of the ship, and hence a higher'thrust deduction fraction,due to propeller action when the propeller is fitted at the bow than when it is fitted at the stern. Therefore, a propeller at the stern results in a higher hull efficiency than a propeller at the .bow. The open water propeller efficiency, the relative rotative efficiency and the hull efficiency are the components of the propulsive efficiency TJD, as is e~sily shown:


TJD

PE

-

PD

109

PE PT PTO PD~ -----

=

PT PTO PD~ PD

PEPTO PDO) - ( -P T- = PT PD~

PTO PD

(5.20)

= TJH TJo TJR

Example 4 A ship has a speed of 18.0knot!'. when its engine has a, brake power of 10000 kW

at 150 rpm. The engine is directly connected to the propeller which has a diameter of 6.0 m'. The effective power of the ship is 6700 kW and the propeller produces a thrust of 900 kN. The open water characteristics of the propeller are, given by:

K T = 0.319 - 0.527 J

= 0.354 -

lOKQ

0.5:78 J

+ 0.169 J2

+ 0.203 J2 .

Determine the propulsive efficiency and its components based on thrust identity. The shafting efficiency is' 0.970. ~

\,

.r.L

v

= 18.0 k

D

=

_

6.0m

= 9.2592 ms- l PE

::::

PB

=

10000kW

T

'=

900kN·

6700kW PE

=

6700 9.2592

RT

::::

PD

= PBTls =

V

T pn 2 D4

J{TB

=

Q

=

21rn

J{QB

=

pn 2 D5

PD

::::

Q

=

=

=

0.970

. ,:: : ;

= 9700kW

900 1.025 x 2.5 2 x 6.0 4

9700 21r x 2.5

Tis

= 150 rpm = 2.5 S-1

723.605kN

10000 x 0.970

=

n

= 0.1084

= 617.521kNm·

617.521 1.025 x 2.5 2 x 6.0 5

= 0.01240

~ ~ ". ,


110 Thrust identity: KT

= KTB

that is: 0.319 - 0.527 J

+ 0.169 J2 = 0.1084

J2 - 3.1183J + 1.2462 = 0 10 KQ = 0.354 - 0.578 J

VA

+ 0.203 J2

= JnD = 0.4707 x 2.5 x 6.0 VA V

1-w =

7.0599

T

= .1 -

t

~~

= 0.1269

= 7.0599ms- 1

= 0.2375

w

= 9.2592 = 0.7625

1_t=RT= 723.605 900 flH

or J = 0.4707

= 0.8040

t

= 0.1960

0.8040 ' 1.0544

0.7625 =

=

0.01269

= 1.0234 0.01240

fiR =

0.1084 0.4707 0.01269 x ~ = 0.6399

= 1.0544 x 0.6399 x 1.0234

flD = flH 110 11R

= 0.6905

= Ps = 6700 = 0.6907

PD . 9700

Example 5 This is the same as Example 4 except that the propulsive efficiency and its compo nents are to be determined by torque identity. From the previous example:

KTB

Torque identity:

= 0.1084

lOKQ

=

J(QB

10KQB

that is:

0.354 - 0.578 J

=

0.01240

= 0.1240

+ 0.203 J2 = 0.1240

. i


111

+ 1.1330

J2 - 2.8473 J

K T = 0.319 - 0.527 J

VA = JnD 1- w I-t

= 0

or J

+ 0.169 J2

= 0.4783 x 2.5 x 6.0

VA =V =

7.1738 = 0.7748 9.2592

RT

723.605 = 0.8040 900 0.8040 = 1.0377 0.7748

=

T

1-t

= =

= 0.4783

.

= 0.1056

= 7.1738ms- 1 w = 0.2252 t

= 0.1960

71H

=

l-w

71R

=

K TB KT -

710

=

KT J K Q 211'

71D

= TJH TJo TJR = 1.0377 X 0.6483 x 1.0265 = 0.6906 =

0.1084 0.1056

=

= 1.0265

0.1056 0.01240

PE = 6700 PD 9700

0.4783 211'

x--

= 0.6483

= 0.6907.

From these examples, it may be seen that there are small diff~rences be tween the individual components of propulsive efficiency based on thrust identity and those based on torque identity. However, the thrust deduction . fraction is the same, and so is the propulsive efficiency, the small differences being due to round-off errors.

5.7

Estimation of Propulsion Factors

The propulsion factors - wake fraction, thrust deduction fraction, relative rotative efficiency and open water efficiency - are often determined with the help of model experiments as described in ChapterS. However, it is usually necessary to have an estimate of the wake fraction, the thrust deduction fraction and the relative rotative efficiency for designing a propeller, and for this a number of empirical formulas and diagrams, based on statistical analyses of model data and ship trials data, are available. Some of these formulas are given in Appendix 4.

ioJ,;

.


112

Problems 1. A ship has a r,esistance of 550 kN at a speed of 16.0 knots with its propelling machinery developing 6000 kW brake power at 120 rpm. The transmission losses are 2 percent. The wake fraction is 0.250, the thrust deduction fraction 0.200 and the relative rotative efficiency 1.030 based on thrust identity. Cala culate the effective power, the thrust power and the delivered power, and the thrust, torque and open water efficiency of the propeller. What is the overall propulsive efficiency? 2.

A ship has a resistance of 500 kN at a speed of 16.0 knots with the engine producing a brake power of 6000 kW at 120 rpm. The propeller thrust is 600 kN and the loss of power in transmission from the engine to the propeller is 180kW. In order to produce the same thrust at the same rpm in open water, the propeller would have to advance ata speed of 12.0 knots and require a torque of 500 kN m. Calculate the effective power, the thrust power and the delivered power, as -well as the hull efficiency, the relative rotative efficiency, the open water efficiency and the propulsive efficiency (quasi-propulsive coef ficient).

3. A ship has a speed of 20.0 knots when the propeller rpm is 180 and the b,rake power of the engine is 15000 kW. The effective power of the ship at 20.0 knots is 10000kW, and the wake fraction is 0.200, the thrust deduction fraction 0.120 and the relative rotative efficiency 1.050, 'based on thrust identity. The shafting efficiency is 0.970. Calculate the delivered power and thrust power, the hull efficiency, propeller open water efficiency and propulsive efficiency, and: the propeller thrust and torque.

4:-' A ship moving

at a speed of 19.5 knots has a propeller directly connected to . , a diesel engine developing 7500 kW brake power at 180 rpm. The shafting efficiency is 0.970, the open water efficiency of the propeller is 0.650, the wake - fraction is 0.250, the thrust deduction fraction 0.200 and the relative rotative . efficiency 1.050. Calculate the effective power of the ship and the propeller thrust and torque:

5. A ship with a propeller of diameter 5.0 m has a speed of 11.66 knots with the propeller running at 90 rpm. The propulsion factors based on torque identity are: wake fraction 0.250, thrust deduction .fraction 0.190, relative rotative efficiency 1.060 and shafting efficiency 0.970. The open water characteristics . of th~ propeller are as follows:

o

0.200

00400

0.600

0.800

0.342

0.288

0.215

0.124

0.028

0.402

0.350

0.276

0.195

0.108


113

Determine the brake power of the engine and the effective power of the ship at this speed. 6.

A twin screw ship has a design speed of 18.0 knots. Its propellers have a diameter of 3.0 m and are designed to operate at an advance coefficient J = 0.7. the corresponding open water thrust and torque coefficients being: KT = 0.350, 10 KQ = 0.560. The propulsion factors are: w = 0.050, t = 0.060, 1/R = 0.980, 1/s = 0.960. Determine the design propeller rpm and the brake power of each of the two engines. What is the effective power of the ship at the design speed?

7.

A ship has a speed of 17.0 knots when its propeller of 4.0m diameter has an rpm of 150 and the engine has a brake power of 4850 kW, the shafting efficiency being 0.970. The effective power of the ship at 17.0knots is 3660kW. If the' propeller operates at the following open water characteristics J = 0.650, K T = 0.300, 10 K Q = 0.475, determine the wake fraction, the thrust deduction fraction, the relative rotative efficiency and the open water efficiency for thrust identity.

8.

A ship with a resistance of 500 leN at a speed of 15.0 knots has a propeller of 5.0 m diameter who,se thrust is 585 kN and torque 425 leN m. The propeller rpm is 120 and the shaft losses are 3 percent. It is estimated that a model propeller of 20 cm diameter when run in open water would produce a thrust of 312.381 N with a torque of 955.317N cm in fresh water at 1800 rpm and 3.704 m per sec speed of advance. Use these data to determine (a) the effective power, the thrust power, the delivered power ~nd the brake power of the ship, (b) the wake fraction and the thrust deduction fraction, and (c) the hull efficiency, the propeller open water efficiency and the relative rotative efficiency. What is the overall propulsive efficiency?

9.

A ship has a propeller of diameter 5.0 m whose open water characteristics are as follows:

J

o

0.100

0.200

0.300

0.400

0.500

0.600

0.700

KT

0.300

0.275

0.243

0.207

0.167

0.125

0.081

0.031

10KQ:

0.315

0.295

0.270

0.240

0.206

0.170

0.128

0.082

The effective power of the ship is 3840 kW at the design speed of 16.0 knots, and the propulsion factors based on thrust identity are: wake fraction 0.200, thrust deduction fraction 0.160, relative rotative efficiency 1.050 and shafting efficiency 0.970. Determine the brake power and the propeller rpm at the design speed. 10.

•

A ship has an effective power of 9000 kW at·a design speed of 18.0 knots. It has twin screws each of 4.0 m diameter connected through reduction gearing


114

to two medium speed diesel engines of brake power 7500 kW each running at 600 rpm. The propellers are designed to operate at J 0.600, K T = 0.150 and 10 KQ 0:230. Determine the gear ratio, the wake fraction and the thrust deduction fraction. The shafting efficiency is 0.950 and the relative rotative efficiency 1.000.

=

\

=

CHAPTER

6

Propeller Cavitation

6.1

The Phenomenon of Cavitation

A liquid such as water begins to vaporise when its pressure becomes equal to the saturation vapour pressure. The vapo/ur pressure of water is 1.704 kN per m 2 at 15°C and 101.325 kN per m 2 (i.e. atmospheric pressure) at 100 °C, the boiling point of water or the temperature at which water evaporates to form steam. If at a point the pressure in water drops to a value equal to the vapour pressure, the water at that point begins to vaporise forming cavities filled with water vapour. The formation of such low pressure vapour filled cavities is called cavitation. A propeller produces its thrust by creating a difference between the pres sures acting on the face and the back of the propeller blades, the pressure on the back of a blade section falling below the ambient pressure and the pres sure on the face rising above it, as shown in Fig.6.1. If the pressure at any point A on the back of the blade falls to the vapour pressure, the water at that point begins' to cavitate. In actual practice, sea water contains minute solid particles in suspension and dissolved gases, and these impurities cause cavitation to start at pressures somewhat higher than the vapour pressure as the solid particles act as nuclei for the formation of cavities and the dis solved gases come out of solution before the water itself starts vaporising. Thus, cavitation in sea water may start when the pressure reaches a value of 17 kN per m 2 (absolute) instead of the actual vapour pressure, which has 115

1_


116 P" V,

-6p Po

-t--7"'"-------t--

+6p·

FACE

FigUl"e 6.1 : Streamlines and Pressure Distdbution ar"ound a Propeller Blade Section.

a val~e of 1.704 kN per m 2 at 15°C for fresh water, the value for sea water being slightly lower. Cavitation in marine propellers usually first manifests itself by an increase in the propeller rpm without a commensurate increase in the speed of the ship. This "racing" of propellers was first noted by Osborne Reynolds in 1873 and was later recorded during the trials of the British torpedo boat "Daring" in 1894. This vessel failed to attain its specified speed with the propellers fitted initially, but when these were replaced by propellers of a greater blade area the speed attained was considerably in excess of that specified. Propeller cavitation also played a part in the trials of the vessel "Turbinia" used by


117

Sir Charles Parsons to demonstrate the steam turbine. The initial trials of the "Turbinia" with a single propeller proved to be disappointing but when three shafts each carrying three tandem propellers were fitted to the ship, speeds in excess of 34 knots were achieved. In the cases of both the "Daring" and the "'Thrbinia", the improved performance was shown to be due to the elimination of cavitation by decreasing the loading on the propellers. The condition for cavitation to occur at a point A on a propeller blade, Fig. 6.1. may be obtained as follows. Let the pressure and velocity at A be PI and VI and the pressure and velocity at a point at the same depth far ahead of the propeller be Po and VO, the velocities being measured with respect to the propeller. Then, by the Bernoulli theorem: PI

+ "2I P V2I

_ - Po

+ "2I P v,2a

so that the difference between the pressures at the point A and at the point far ahead is: A L.l

P :;::

PI -

_

I

TT2

I

TT2

Po - "2 P va - 2 P VI

or dividing by the "stagnation pressure" q = ~ PV{ to make the equation non-dimensional:

D.. P == PI - Po :;:: 1 ~ (VI) '2 q ~ P "'02 Va If cavitation starts at the point A, then PI :;:: PV so that the condition for cavitation to occur is: '

D..p q

(6.1)

=

If the pressure Po is taken as the total static pressure (atmospheric plus hydrostatic pressure) at the point A and Va as the relative velocity of water then the condition for cavitation to occur may be written as:

D..p

Po -Pv

q

.!'2 PVa2

(6.2)


118

where 0A is the "local" cavitation number at A. The quantity 6.p/q is sometimes written as the pressure coefficient Cp , so that the condition for cavitation to occur on a propeller blade section is that the minimum value of C p be equal to the local cavitation number. This is normally written as follows:

(6.3)

For a blade section at a non-dimensional radius x = r / R and angle 0 to the upwardly directed vertic.alline ("12 o'clock position"),

-Cpmin =

PA + Pg (h - x R cos 0) - PV ~P [Vl + {211'nxR - "'tPl

(6.4)

where Va is the axial component of the velocity and "'t the tangential com ponent (taken positive in the direction of motion of the blade) at the; point I (x, 0), h being the depth of immersion of the propeller shaft axis. i Example 1 A propeller blade section begins to cavitate when its relative velocity with respect to undisturbed water is 32 m per sec and its depth below the surface of water is 4.0 m. Determine the velocity of water with respect to the blade at the point where

cavitation occurs, assuming that cavitation occurs when the local pressure falls to the vapour pressure.

Vo = 32.0ms- 1

h = 4.0m

PA = 101.325kN.m- 2

PV

= 1.704kNm- 2

Po = PA + p gh = 101.325 + 1.025 x 9.81 x 4.0 == 141.546 kN m- 2

q == Ap q

Vi

=

~pvi = ~ x 1.025 PO-PV 1

:'i.P

=

°

V;2

1.2665°·5

= X

X

32.0 2 :::f 524.8kNm- 2 .

141.546 - 1. 704 = 0.2665 = 524.8

(~r -1

Vo = 1.1254 x 32.0 = 36.012ms- 1

I ~


6.2

119

Cavitation Number

As indicated in Section 4.4, the cavitation number U of a propeller is often defined by taking Po as the total static pressure at the propeller axis and va as the speed of advance "A.: U

= PA +:g\- PV

(6.5)

2 PVA where P.4. is the atmospheric pressure and h the depth of imT!lersion of the propeller axis.

Instead of the speed of advance, one may take, the propeller blade tip speed -rrnD as Vo:

= PAl + pgh -

Un

PV

(6.6)

2P(-rrnI?)2 or even the resultant of the speed of advance and the tip speed: UR

=

PA +pgh - PV

~ p [Vl

+ (-rrnD)2]

(6.7)

The propeller blade section at 0.7R is often taken as representing the whole propeller, and the cavitation number is then defined in terms of the pressure at the shaft axis and the relative velocity of the blade section at 0.7R with respect to undisturbed water (induced velocities being neglected):

uo. 7R =

PA +pgh -PV

"""1-----=---'-~----"-----

2" P

[vl + (0.7 -rrnD)2]

(6.8)

On the other hand, it has been observed that in many propellers it is the blade section at 0.8R that is most susceptible to cavitation, and since cavitation is most likely to occur when the blade section is at its minimum depth of immersion, the cavitation number should be defined in terms of the pressure and relative velocity in this condition: PA
+ pg(h -

~p

[vl +

O.~R)

- PV (0.8 -rr n D)2]

(6.9)

1 Basic Ship Propulsion

120

With so many different definitions of the cavitation number of a propeller, it is essential to state clearly the definition being used in a particular ap plication. The widely different values obtained for the cavitation numbers defined in different ways are shown by the following example.

Example

2

A propeller of 6.0 m diameter has a speed of advance of 8.0 m per sec and an rpm of 108, its axis being 5.0 m below the surface of water. Calculate the different cavitation numbers.

n

D = 6.0m

= 108 rpm = 1.8 S-l

h = 5.0in

Cavitation number based on pressure at the shaft axis and the speed of advance:

0"

=

+ pgh -

PA

PV

12 pV'2 A

=

101.325 + 1.025 x 9.81 x 5.0 - 1. 704 ~ x 1.025 X 8.0 2

Cavitation number based on blade tip speed: PA

+ pgh -

PV

~p(7I"nD)2

=

101.325 + 1.025 x 9.81 x 5.0 - 1.704 = 0.2541 x 1.025 X (71" X 1.8 x 6.0F

!

I ~

Cavitation number based on the resultant velocity at blade tip:

I

O"R

=

101.325 + 1.025 x 9.81 x 5.0 - 1.704 + pgh - PV = ! x 1.025 [8.02 +(; x 1.8 X 6.0)2] = ~ p[vl + (rrnD)2] PA

, I

0.2407

I

J

Cavitation number based on the resultant velocity at 0.7R: - PV = = ~ p[Vl++pgh (0.711"nD)2] PA

O"O.7R

=

0.4657

- - - - - - - - - - - - _ . ----_._

101.325 + 1.025 x 9.81 x 5.0 - 1. 704 ~ x 1.025 [8.0 2 + (0.711" x 1.8 x 6.0)21

I

I ~

• I 1

i

121

Propeller Cavitation Cavitation number based on the pressure and relative velocity at 0.8R:

+ pg(h - 0.8R) - p'/ ! p [V~ + (0.87r n D)2]

PA O"O.8R :::=

6.3

=

101.325 + 1.025 x 9.81 (5.0 - 0.8 x 3.0) - 1.704 ~ x 1.025 [8.0 2 + (0.8 x 7r x 1.8 X 6.0)2]

:::=

0.3065

Types of Propeller Cavitation

Cavitation in a propeller may be classified according to the region on the propeller where it occurs, viz. tip cavitation, root cavitation, boss or hub cavitation, leading edge cavitation, trailing edge cavitation, face cavitation and back cavitation. Cavitation at a particular location of the propeller indicates a region of low pressures and high velocities. It is often possible to reduce or eliminate such local cavitation by making suitable changes in the propeller geometry, e.g. reducing the pitch and increasing the blade width locally. I

Cavitation may also be classified according to the nature of the cavities or their appearance: sheet cavitation, spot cavitation, streak cavitation, cloud cavitation, bubble cavitation and vortex cavitation. In sheet cavitation, the cavity is in the form of a thin sheet cove,ring a large part of the propeller blade surface. Spot cavitation occurs at isolated spots on an uneven blade surface where rough spots cause localised pressure drops. A large number of cavitation spots close toge!hl::~_~~y.]'~tUljn ..acavltyjn.the form of astre~k. Cloud cavitation usually occurs at the end of asheetc~vity when it disinte grates to form a large number of very small cavities having the appearance of a cloud. In bubble cavitation, spherical cavities are formed at points where the local pressure approa.ches the vapour pressure. These bubbles grow in size as they move downstream into a region where the pressure would theo retically fall had there been no cavitation. When these vapour filled bubbles then move into a region of pressure which is higher than the vapour pres sure the vapour condenses and the bubbl0s c.ollapse, often suddenly and with tremendous force in a process called implosion (the opposite of explosion). A

122


'I

','

propeller blade producing thrust sheds vortices from its trailing edge, these vortices being particularly strong at the tip and the root. The pressure at the core of a vortex is lower than that in the outer layers, and if the pressure at the vortex core falls to the vapour pressure vortex cavitation results.

r

~,

"t'

Sheet cavitation usually begins at the leading edge of a propeller blade when blade sections work at large ~ngles of attack causing a sharp negative pressure 'peak to occur close to the leading edge. If the angle of attack is positive sheet cavitation occurs on the back of the blade, whereas if the angle of attack has a large negative value sheet cavitation occurs on the propeller blade face. Blade sections that .work at zero ("shock free entry") or small angles of attack usually do not suffer from sheet cavitation unless the sections are specially designed to promote sheet cavitation as in "supercavitating" propellers. (Such propellers are discussed in Chapter 12.) Bubble cavitation occurs in propellers which have aerofoil sections not specially designed for a uniform pressure distribution over the back. Bub bles are formed just upstream of the position of maximum thickness where the pressure falls close to the vapour pressure, grow larger as they, move downstream, and collapse violently shortly after they reach the poinV.where the pressure rises above the vapour pressure. The impact of the collapsing bubble is extremely high, and stresses which may be as high as 2800N per mm2 are said to be generated. The repeated collapse of these bubble cavities on the propeller blade surface causes rapid erosion of the blade eventually causing it to break. Cavitation erosion due to bubble cavitation is a seri ous problem in heavily loaded propellers. Cloud cavitation is also similarly harmfuL Vortex cavitation occurs in the vortices that are shed from the propeller blade tips and roots. The strength of the vortex shed from the blade tip increases downstream as the induced velocities increase, and cavitation in the tip vortex begins some distance downstream of the propeller. This type of cavitation is called unattached tip vortex cavitation. An increase in the loading of the propeller causes the vortex cavity to move upstream gradually until it begins at the blade tip, resulting in attached tip vortex cavitation. A further increase in propeller loading causes the cavity to enlarge at the blade and spread progressively in the form of a sheet from the tip to lower radii on the back of the blade. Cavitation in the vortices shed by the roots of the propeller blades has the appearance of a thick rope consisting of a number

,

4

~

f

I ~

I "r

J.

I

! I I

'i

j,

I

I

'1'~

'f i

;f>,

J ;

{

,II , \


123

of strands, one for each blade of the propeller. Sometimes, a vortex cavity extends from the propeller to the hull of the ship; this is called propeller hull vortex cavitation. Fig. 6.2 depicts the various types of propeller cavitation. The occurrence of the different types of cavitation on a propeller depends upon the cavitation

/~

"

UNATTACHED

TIP

ATTACHED

VORTEX

. SPREADING

CAVITATION

I.

Pv -i----+---\-

- .6p

HUB VORTEX CAVITATION

Po -l=7~===---.--c:::::l +6p 8UBBLE CAVITATION

SHEET AND CLOUD CAVITATION

FACE CAVITATION

SPOT AND STREAK CAVITATION

Figure 6.2: Types of Propeller Cavitation.

..J...

_

.


124

number (J and the advance coefficient J, and this is illustrated in Fig. 6.3 by a type of diagram due to R. N. Newton (1961). TIP VORTEX

ATTACHED

UNATTACHED

PROPElLER OPERATING

RANGE

CAVITAnON FREE ZONE

/ I

(J

.

J Figure 6.3: Occurrence ofDifferent Types of Propeller Cavitation.

6.4

Effects of Cavitation

Cavitation affects the nature of the flow around a propeller since the flow is no longerllOmogeneous. The formation of cavities has the effect of virtually alt~;ing the sh~.pe t'he propeller blade sections, and as a result the thrust and, to a lesser extent, the torque of the propeller are reduced, and so also the propeller efficiency. The effect of cavitation on the open water characteristics of a propeller is shown in Fig. 6.4. The result is that increased power is required to aUain a given speed, and in cases of severe cavitation the ship may not achieve the specified speed.

of

Cavitation can also cause serious damage to a propeller, and sometimes, to a rudder placed in the propeller slipstream. As indicated earlier, the collapse

--~---

------' .. _

,


125

J

Figure 6.4. : Effect of Cavitation on Propeller Performance.

of bubble cavities results in very high impact pressures and the repeated collapse of such bubbles at a particular location of the propeller blade can cause rapid erosion of the blade leading ;to its brea:king off. If these bubbles collapse near the blade tip or the trailing edge where the blade section is thin, the resulting impact pressures may cause the blades to ben~. Sheet cavities and vortex cavities usually disintegrate into clouds of very small bubbles and the collapse of these bubbles on the propeller blades may also damage the propeller. The bubbles in cloud cavitation may be carried to the rudder placed behind the propeller, and adversely affect its performance due to the disruption of flow around it. If these bubbles collapse on the rudder, the rudder surface may be damaged due to cavitation erosion. Corrosion and erosion tend to reinforce each other since the roughened spots created by corrosion promote cavitation, and the pitting produced by cavitation erosion provides a site for corrosion attack. Another important effect of propeller cavitation is vib~~Jgn _.an
.J.

_

126


underwater noise is particularly unacceptable in warships, whose propellers have therefore to be designed to be free of cavitation over their complete operating range.

6.5

Prevention of Cavitation

Owing to the detrimental effects of cavitation, propellers are normally de signed so that they do not cavitate in their operating conditions, or at least so that cavitation is restricted to a level at which its effects are negligi ble. In the case of small high speed craft, however, the propeller operating conditions - very high speeds and powers, high rpm, restricted diameter are sometimes such that avoiding cavitation is virtually impossible, and the propellers are then designed to operate in the fully cavitating regime. Propeller cavitation can be reduced or eliminated basically by three meth ods: (i) Increasing the cavitation number,

(ii) Decreasing the loading on the propeller, (iii) Designing the propeller for uniform loading.. As ;may be observed from Fig. 6.4, a lower cavitation number is associated with increasing cavitation. It follows, therefore, that a higher cavitation number will reduce cavitation. The cavitation number may be increased by increasing the depth of immersion of the propeller, and also by decreasing the relative velocities of the propell~r blade sections, Le. decreasing the speed of advance and rpm of the propeller. Generally, 'however, these variables are determined by other considerations, and increasing the cavitation number to reduce cavitation is an option that is rarely available; The loading on a propeller is usually taken to mean the thrust per unit projected blade area, TjAp. Therefore, if it is anticipated that a propeller is likely to cavitate,a possible solution to the problem is to increase the blade area - the solution adopted in the case of the "Daring". The loading may, of course, also be reduced by distributing the load among a larger number of propellers, which is what was done for the "Thrbinia". Decreasing the


127

speed of the ship and hence the thrust of the propeller would also reduce cavitation, but this may not be an acceptable design solution. Designing a propeller to have uniform loading requires that the propeller blade sections be so selected that the pressure distribution on the back of a blade section is as constant as possible over the chord. It has been observed that uniform suction (Le. uniform negative pressure with respect to the am bient pressure) on a propeller blade section is more likely to be achieved if the blade section has a "shock free entry" 1 Le. the incident velocity is tan gential to the camber centre line of the section or the angle of attack is zero, the centre line has a uniform curvature, and the face and back of the section also ~ve a uniform curvature. Thus, segmental sections (as used in the Gawn Series for example) are less prone to cavitation than aerofoil sections (as used at the inner radii in the B-Series). A popular section shape for propeller blades in which uniform suction is required is the Karman-Trefftz section consisting of two ci;rcular arcs. Fig. 6.5 shows the streamlines around

i·A·.-:-~

---

:----.

BACK

bap q

-

, t

+

I

+

\

FACE

Figure 6.5: Uniform Suction Section.

Basic Ship Propulsion .

128

such a section and the pressure distribution on it. One may compare this figure with Fig.6.1. Other blade sections widely used in heavily loaded pro pellers are the NACA-16 and NACA-66 sections with a = 0.8 and a = 1.0 mean lines, in which the pressures at the back of the section are uniform over 80 percent and 100 percent of the chord respectively. Details of these section shapes are given in Appendix 2. It is difficult to ayoid cavitation when a heavily loaded propeller wC!rks in a non-uniform wake since the incident velocity and the angle of attack for any radial section fluctuate over a wide range depending upon how much the wake velocity "aries at that radius. Reducing cavitation therefore also requires the hull form to be designed and the propeller located so that it works in as uniform an inflow as possible.

6.6

Cavitation Criteria

Following the trials of the "Daring" and the "'I'urbinia" , it was suggested that cavitation in a propeller could be avoided if its thrust loading was limited to a figure such as 11.25lbs per square inch of projected area (77.57 kN per m 2 ). Later on, it was felt that the propeller blade tip speed was a better criterion for preventing propeller cavitation, a recommended value. being 12000 ft per min (about 61 m per sec). l'vlore modern concepts about the mechanism of cavitation and data based on the performance of propp-llers in ships as well model tests have led to cavitation criteria based on a non-dimensional form of the thrust loading and the cavitation number. One of the most widely used cavitation criteria for marine propellers is a diagram introduced by Burrill (1943), Fig. 6.6. The diagram gives the limiting value of a thrust loading coefficient 'fa as a function of the cavitation number
(i) Warship propellers with special sections, (ii) Merchant ship propellers (aerofoil sections), (iii) Tug and trawler propellers,


129

0.50 0.40 0.30

I. UPPER UI.IIT LE\'£L rOR HICHLY LOAOEO PROPELLERS (WARSHIP PROPELLERS WIlli SPECIAL SEC nONS)

-

I-I-

2. SUGGESTED UPPER LIMIT rOR MERCHANT SHIP PROPELLERS .3. SUGGESTED UPPER UI.IIT fOR ruG AND ffiAYltER PROPEU.ERS

.. [,.-1--

l3- v

V

0.20

+- 1 17"0.15

1-0

~

3

. GO'

t-- 1--' 1-

/

L'"

/ V

0.10

-

.- -- .- _.,

0.05 0.05

'- f- ' - -

-. -

- ..

._.

1/ 17 1,

..

.

_..--

··1 -

0.10

-" ._.

I -I- .-

...

0.15

0.20

-_. -

.

.•

_.. -.-. _. ..- ~

~- 1·-

-_.

0.30 0.40 0.50

1.00

-

-.

- - ... -

'.

1.50 2.00

Figure 6.6: Burrill Cavitation Diagram.

where:

T

7C

= ! p [Vi +AP (0.77r n D)2]

(6.10)

The projected blade area Ap is given approximately by: Ap = (1.067 - 0.229

~) AD

(6.11)

where AD is the developed blade area, which for practical purposes may be taken to be equal to the expanded blade area. It has been suggested thai for propellers having elliptical blade outlines, this formula should be modified to: Ap

=

(1.082 - 0.229

~) AD

(6.12)

Experimental data subsequently published qy Burrill and Emerson (1962) showed that the limiting line for warship propp-llers lies close to a line

J..

_


130

representing 10 percent back cavitation, while the limiting line for merchant ship propellers lies close to a line rep~esenting 5 percent back cavitation. The three lines in: the Burrill diagram can be represented quite accurately by the following equations: (i) TC

Warship propellers with, special sections:

=

0.0130 + 0.5284 O'O.7R + 0.3285 0'~.7R -1.0204 0'5.7R (0.11 ~

(ii) TC

TC

(6.13)

0.43)

Merchant ship propellers with aerofoil sections:

=

,

0.0321

+ 0.3886 •O'O.7R -

(0.12 ~ (iii)

O'O.7R ~

2

3

0.1984 O'O.7R + 0.0501 O'O.7R (6.14)

O'O.7R

$ 1.50)

Tug and trawler propellers:

= 0.0416 + 0.2893 O'O.7R - 0.1756 0'5.7R + 0.0466 0"5.7R (6.15) (0.28 ~

O'O.7R ~

1.60)

These equations should not be used outside the ranges specified. 'Another criterion which may be used to determine the expanded blade area required to avoid cavitation is due to Keller (1966): AE _

Ao

(1.3

+ 0.3 Z) T + k

(po - pv) D2

(6.16)

where k is a constant, equal to 0 for high speed twin screw ships such as naval vessels with transom sterns, 0.1 for twin screw ships of moderate speed with cruiser sterns and 0.2 for single screw ships. Although cavitation-free propellers have been successfully designed for decades using simple cavitation criteria such as those due to Burrill. and to Keller, lt must be realised that cavitation depends "'---'~ not merely on the •.._-. thrust loading and the cavitation number, but also on the non-uniformity of .-....---.~._._._ ..

_----_._-_

...

.,


131

6r-T-r-,-,r-r-rTlr-r-rTl-r""-::;r-"lr::;::s>"'-T:::!',...,..."r-r::;pooTO-r.,..-,

5 4

.3

A

2

DEGREES d...

O~~mf11::::=:~~::!::~=~~~~~+;;:-~:=J 0.02

-1

~

- .• - - •. 0.04 -. - - - - - 0.06 0.08

-2

0.,/"0

-.3

.'

-----I

0

-4

r i

- 5 .......-'-..I...-''--'--'-.L.-J'-'--'-"''-'-'''--'-'''''-''-'-''''"-.L...l.....L.:'''-''--'--'--'-'-c=-.J 0.0 0.2 0.4 0.60.8 1.0 q 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 .3.0

-cPmin Figure 6.7: Minimum Pressure Envelopes.

wake and the detailed geometry of the propeller blade sections. Cavitation characteristics of aerofoil sections have therefore been dev~rmined as a func tion of the thickness-chord ratio and the angle of attack for different camber -----. ratios and thickness distributions. A typical diagram of this type, as shown in Fig. 6.7, may be obtai~~~i9.r.l;l...s~st.i,
The following examples illustrate the use of these cavitation criteria. Example 3 A propeller of diameter 5.5 m and pitch ratio 1.0 has its axis 4.0 m below the water line. The propeller has a speed of advance of 7.0 m per sec when running at 120 rpm I

r

1_·__


132

and produces a thrust of 520 kN. Determine the expanded blade area ratio of the propeller using the Burrill criterion for merchant ship propellers.

P

D

=

5.50m

n

=

120 rpm

(JO.7R

= TO

PA

=

T = 520kN

2.0 S:-l

+ pgh ~

h = 4.0m

D = 1.0

PV·

P V6:7R

101.325 + (1.025 X 9.81 x 4.0) - 1.704 = 0.4303 ~ x 1.025 x 634.169

=

· 2 3 = 0.0321 + 0..3886 O"O.7R - 0.1984 (JO.7R + 0.0501 O"O.7R

520

~

Ap

AP x 1.025 x 634.169

=

1.5999 0.1666

Ap 1.067 - 0.229 P / D \

AE Ao

=

=

=

1.5999 Ap

'Ir

0.1666

9.6032m2

9.6032 1.067 - 0.229 x 1.0

=

=

= 0.1666

11.460 x 5.502 /4

=

= 11.460m2

0.4824

Example 4 A twin screw high speed ship with a transom stern has three-bladed propellers of 3.0 m diameter, the propeller axis being 3.5 m below the water surface. The ship has a speed of 30 knots at which its effective power is 10000 kW and the thrust deduction fraction is 0.06. Determine the expanded blade area ratio using the Keller criterion.

No. of prop'ellers

v =

=

2

30 k = 15.432 ms- 1

D

= 3.0 m

Z t

= =

3 h 0.06

= 3.5 m

PE

10000kW

Propeller Cavitation Po - pv

133

= PA + pgh -

= 101.325 + 1.025 x 9.81 x 3.5-1.704

PV

= 134.814 kN m- 2

PE

=

R

=

T

1 R -=2 1- t

AE Ao

=

V

10000 15.432 ::::

=

648.004kN

648.004 2(1 - 0.06)

(1.3 + 0.3 Z) T (pO - pv) D2

+k

=

=

344.683kN

(1.3 + 0.3x 3) x 344.683 134.814 X 3.02

+0

= 0.6250

Example 5

In a t\\in screw ship, the propellers have a diameter of 3.0m. The pitch ratio is 0.670 at 0.7R. The propeller centre line is 2.0m below th.e waterline. The blade section at 0.7 R is of an aerofoil shape for which the limiting angles of attack to avoid sheet ca\-itation on the face and the back are given respectively by amin = -0.40 0.60 (-Cpm1n ) and a max = 1.35 + 0.65 (-Cpm1n ), a in degrees. The propellers run at 120 rpm. The axial and tangential velocity components at 0.7 R when a propeller blade is in the vertically upward position are 3.50 m per sec and 0 respectively. When the blade makes an angle of 120 degrees with its vertically upward position, the axial velocity component is 5.05 m per sec and the tangential velocity component directed opposite to the revolution of the blade is 2.50 m per sec. If these blade positions result in the maximum and minimum angles of attack of the blade section at 0.7R, show that it operates without cavitation.

D = 3.00m

P

D = 0.670

h = 2.00m

r x = R

=

0.7

n = 120rpm Limiting values of a :

amin

= -0.40 - 0.60 (-Cpmin

ama..'C

= '1.35 + 0.65 (-C/Jm1n

)

)


134

e=

0

tan

0.670 . = 0.3047 x 0.7

'Vt =

0

'Vt =

-2.5 ms- 1

1r

() = 0: =

+ pg(h- xR cos()) - PV kp [V; + (21rnx,R-vt)2]

P.4.

= 101.325 + 1.02~ x 9.81 (2.0 - 0.7 x 1.5 x cos 0) - 1.704!

~

x 1.025 [3.5 2 + (2 x

1r

x 2.0 x

0~7 x

I

1.5 - 0)2]

= 1.1431

Limiting values of cx :

tan(3 =

. Va 21rnxR -

'Vt

=

2x

CXmin

= -0040 - 0.60 (1.1431) = -1.086°

CX max

= 1.35 + 0.65 (i,1431)

1r X

= 2.093°

3.5 = 0.2653 2.0 x 0.7 x 1.5 - 0

. (3 = 14.856°

\

CX

=,

= 16.944 - 14.856 = 2.088° : within the limit for back cavitation.

101.325 + 1.025 x 9.81 (2.0 - 0.7 x 1.5 x cos 120°) - 1.704 = ~ x 1.025 [5.05 2 + {2 x 1r ~ 2.0 x 0.7 x 1.5 - (-2.5)}2]

= 0.8974 Limiting values of CX

:

Qmin

= -0040 - 0.60 (0.8974) = -0.938°

a max = 1.35 + 0.65 (0.8974) = 1.933°


tan/3

0:

=

2 x 11"

135

X

5.05 2.0 x 0.7 x 1.5 + 2.5

= Y - /3 = 16.944 -17.836

= -0.892° :

= 0.3218

/3 =

17.836°

within the limit for face cavitation.

The minimum and maximum angles are within the limiting values of angle of attack, and hence there is no cavitation.

6.7

Pressure Distribution on a' Blade Section /

The pre~sure distribution on the face and back of a propeller blade may be determined through the lifting surface th~ory. The pressure distribution on an individual blade section may also be obtained by conformal mapping techniques. However, a simple approximate method has been found to give sufficiently accurate res]llts and is therefqre widely used to determine the pressure distribution over aerofoil sections and propeller blade sections. In this method, which is due to Allen (1939) and is described in Abbott and Doenhoff (1959), the increase in velocity ~ V VI -'Va at any point on the surface of a blade section (see Fig. 6.1) compared to the undisturbed velocity Vo is taken to be due to three causes: (i) the blade section camber, (ii) the thickness distribution and (iii) the angle of attack.

=

The increase in velocity ~ Vf due to camber is given for the "ideal" angle of attack Qi (the angle of attack for which the most favourable pressure distribution on the back of the blade section is obtained), the camber ratio f Ie being such that the lift coefficient CLi at the ideal angle of attack is 1.0. D.Vf is proportional to the camber ratio, so that : (6.17) where ~ Vn is the increase in velocity due to camber for CLi = 1.0 and (f I ch is the corresponding camber ratio. Values of D. Vfl , for some blade sections used in marine propellers are given in Appendix 5. The increase in velocity D. Vi due to the ,thickness distribution ("fairing shape") depends on the thickness-chord ratio tic for a given section shape. I ~/·

(

1_______


136

Values of A vt for some blade sections used in marine propellers are given in Appendix5. The increase in velocity D. Va due to the angle of attack a depends on the thickness-chord ratio of the blade section and the angle of attack measured with respect to the ideal angle of attack ai. The values of A Va; corresponding to the ideal angle of attack and some standard thickness-chord ratios are given for different mean lines. These values must be first corrected to the desired thickness-chord ratio by interpolation. The values of D. Va for a giver angle of attack a are then obtained from: (6.18)

a and Qi being in radians.

,'

The increases in velocity due to camber, thickness distribution and angle of attack are added together and the velocity at a point on the blade se,ction obtained: ' ; (6.19)

the positive signs corresponding to the back (suction surface) of the blade section and the negative signs to the face (pressure surface). The change in pressure compared to the l.mdisturbed pressure is then obtained as:

Apq

=

1-

(Vva1)2

(6.20)

This can be calculated for diff~rent positions along the face and back of the section to obtain its pressure distribution.

Problems 1.

A propeller of diameter 4.0 m, pitch ratio 0.8 and blade area ratio 0.50 has its axis 3.0 m below the surface of water. The operating conditions of the. propeller correspond to J = 0.500, ]{T = 0.140. The propeller cavitates when its rpm exceeds 150. Calculate the limiting value of thrust per unit projected blade area and the correspondiI).g cavitation number based on the speed 'of advance.


137

2. A propellerof 6.0 m diameter has an rpm of 150, its axis being 4.5 m below the surface of water. What is the maximum speed of advance that the propeller can have before the blade sections at 0.8R begin to cavitate, given that the maximum relative velocity of water at a point on the section is 7.5 percent greater than the velocity of that point with respect to undisturbed water? Assume that the propeller operates in a uniform wake and that cavitation occurs at the vapour pressure. 3. A ship has a propeller of diameter 5.0 m, pitch ratio 0.8 and expanded blade area ratio 0.750. The ship has a speed of 20 knots with the propeller running at 180 rpm. The effective power of the ship .at this speed is 8000kW, the wake fraction being 0.200 and the thrust deduction fraction 0.120. The upper permissible limit of propeller thrust loading is given by: . . . 7"c

0.6 . = O.27 eTO.7R .

What should be the minimum depth of the propeller shaft axis below the waterline?

4. A ship has a resistance of 677 kN at its design speed of 15 knots. The wake fraction is 0.250 and the thrust deduction fraction 0.200. The propeller of diameter 5.0m and pitch ratio 0.8 has its axis 6.5m below the load water line and has a design rpm of 150. Determin~ its expanded blade area ratio based on the Burrill criterion for merchant ship propellerS. . 5. A propeller of diameter 4.0 m and pitch ratio 0.9 has an expanded blade area ratio of 0.500. The propeller axis is 2.5 m below the surface of water. As the propeller rpm is changed, the speed of the ship changes in such a way that the advance coefficient remains constant: J

= 0.500

Kr = 0.150

10KQ = 0.170

The wake fraction based on thrust identity is 0.250, the thrust deduction 0.200 and the relative rotative efficiency 1.050. Find the speed at which the propeller will begin to cavitate, and the corresponding effective power and delivered power. Use the Burrill criterion for merchant ships. 6. A four bladed propeller of 5.0m diameter and 0.55 expanded blade area ratio in a single screw ship is required to produce a thrust of 500 kN. Determine the minimum depth of immersion of the shaft axis if the propeller is not to cavitate. Use the Keller criterion. 7. A single screw ship has a five-bladed propeller of 6.0 ill diameter and· 0.75 expanded blade area ratio with the propeller axis 5.0 ill below the wa.terline. The effective power of the ship can be approximated by PE = 0.463 Vj}·li with P E in kW and the ship speed VK in knots. The thrust deduction fraction is

138

Basic Ship Propulsion 0.260. Estimate the ship speed at which the propeller will begin to cavitate. Use the Keller criterion.

, ~

'I

8. A single screw ship has a propeller of 6.0 m diameter running at 108 rpm. The blade section at 0.7R has a pitch ratio of 0.8 and a thickness chord ratio of 0.08, for which the cavitation bucket data to avoid face and back cavitation are as follows: 0.40

0.50

0.60

0.70

-0.69

-0.80

-0.91

-1.12

1.55

1. 70

1.85

2.05

fi I

The axial and tangential components of the relative velocity of water at 0.7R

are 7.80m per sec and 0 when the propeller blade is in the vertically tip

position (0 degrees), and 8.25m per sec and 2.00m per sec when the blade is

horizontal(90 degrees). Determine the minimum depths of immersion of, the

propeller shaft axis to avoid back cavitation and face cavitation completely

for these two -blade positions.

9. A twin screw ship has propellers of 4.0 m diameter running at l50rptn, the immersion of the shaft axes being 3.6 m. For the starboard propeller at '0. 75R, the maximum and miniII:\um axial velocity components of water are 10.20 m per sec when the blade is at an angle of 105 degrees to the upward vertical, and 9.40 m per sec at 270 degrees. The tangential velocity components are negligible. The propeller blade section at· 0.75R is an aerofoil of thickness chord ratio 0.06 for which the following cavitation data are given:

CL

= 0.1097

(1 - 0.83 ~) (0:0 + 2.35)

-Cpmll>

0.40

0.50

0.60

0.70

O:~ln

-0.44

-0.59

-0.71

-0.80

o:~ax

1.26

1.41

1.55

1.68

Find the pitch ratio at 0.75R such that there is equal margin against both

face and back cavitation and the range of values of CL'

10. A single screw ship has a propeller of5.0m diameter running at 120rpm. The propeller shaft axis has an immersion of 4.8m. The blade section at 0.7R has a pitch ratio of 0.75 and a thickness chord ratio of 0.07. The cavitation data for the section are as follows:

'1J iI

\

I

, I

,t

139

Propeller Cavitation -Cpm1n

0.40

0.50

0.60

0.70

0.80

0.90

1.00

a~in

-0.75

-0.90

-1.04

-1.17

-1.29

-1.39

-1.48

a~ax

1.16

1.29

1.42

1.54

1.66

1.75

1.86

The longitudinal, transverse and vertical components of the relative velocity of water at 0.7R for the different angular positions of the propeller blades are as follows: ()O

V.,ms- 1 , VII ms- 1 Vzms- 1

o 6.17

o 0.54

30 6.42 -1.86 0.79

60 6.91 -1.60 1.09

90 7.24 -1.94 1.41

120 7.41 -1.72 1.17

150 7.16 -0.96 0.88

180 7.65

o 0.59

(x-axis is positive aft, y-axis positive starboard, z-axis posftive upward, and the propeller is right-handed.) , Plot a as a function of "",Cpm1n and indicate the ~egi~D. in which cavitation in the blade section at 0.7R.

OCc::UfS

\

J-

_

CHAPTER

,

7

<'

{'I

Strel1.gth of Propellers 7.1

Introduction

The propeller is a vital component essential to the safe operation of a ship at sea. It is therefore important to ensure that ship propellers have ade quate strength to withstand the forces that act upon them. On the other hand, providing excessive strength would result in heavier propellers with thicker blades than necessary, leading to a reduction in propeller efficiency. A method is therefore needed to calculate the forces acting on a propeller and the resulting stresses, so that the propeller has just the necessary strength for safe operation in service. The forces that act on a propeller blade arise from the thrust and torque of the propeller and the centrifugal force on each blade caused by its revo lution around the axis. Owing to the somewhat complex shape of propeller blades,. the accurate calculation of the stresses resulting from these forces is extremely difficult. Moreover, while one may be able to estimate the thrust and torque of a propeller with reasonable accuracy for a ship moving ahead at a steady speed in calm water, it is difficult to determine the loading on a propeller when a ship oscillates violently in a seaway and the propeller emerges out of water and then plunges sharply into it at irregular intervals. The effects of the manoeuvring of a ship on the forces acting on the propeller are also difficult to estimate, particularly for extreme manoeuvres such as "crash stops". One must also take into account the fact that even in calm water the forces acting on the propeller blades are not constant but vary dur

140

Strength of Propellers

141

ing each revolution due to the non-uniform wake in which a propeller works. Finally, a propeller must also withstand the effects of the stresses that may be locked into it during its manufacture, of propeller blade vibration and of corrosion and erosion during its service life.

It is thus evident that the accurate determination of propeller strength is an extremely complex problem. In practice, therefore, it is usual to adopt fairly simple procedures based on a number of assumptions to make the problem less intractable, and to allow for the simplifications by ensuring that the nominal stresses determined by these procedures have values which experience has shown to be satisfactory. The ratio of the ultimate tensile strength of a propeller material and the allowable stress (factor of safety or load factor) used in the simplified procedures for determining propeller blade strength is high, often lying between 10 to 2'0. Among the simplifications made in the ,Procedures for determin: peller blade strength are:

~g

pro

I

(i) Each propeller blade is assumed to,be a beam cantilevered to the boss. (ii) The bending moments due to the forces acting on the blade are as sumed to act on a cylindrical section, i.e. a section at a constant radius. (iii) The stresses in the cylindrical section are calculated on the basis of the simple theory of the bending of beams, the neutral axes of the cylindrical section being assumed to be parallel and perpendicular to the chord of the expanded section. (iv) Only the radial distribution of the loading is considered, its distribution along the chord at each radius being ignored. (v) Calculations are carried out only for the ship moving at constant veloc ity in calm water, the effects of manoeuvring, ship motions in a seaway and variable wake not being taken into account. Further simplifications are made in some methods for estimating propeller blade strength.

Basic Sbip PropuLSion

142

7.2

Bending Moments due to Thrust and Torque

Consider a propeller with Z blades and diameter D operating at a speed of advance VA and revolution rate n with a thrust T and a torque Q. The bending moments due to thrust and torque at the propeller blade section at a radius oro may then be determined. . LetdT be the thrust produced by the Z blade elements between the radii rand r + dr, Fig. 7.1. The bending moment due to the thrust on each element at the section ro is then: 1 . dMT = - dT(r - ro)

Z

.

(7.1)

so that the bending "moment at the section due to the thrust on the blade is:

MT

=.

l

R 1

ro

-Z -dT (r dr

ro) dr

(7.2)

The thrust T and the bending moment due to thrust M T act in a plane parallel to the propeller axis. .

.

.

If ~Q is the torque of the Z blade elements between rand r + dr, the force causing this torque on each of these elements in a plane normal to the propell~r axis is dQ/r Z, the resulting bending moment at the section at radius ro being:

(7.3)

I

R

1 dQ (r - ro) dr ro r Z dr

l

II

I

The bending moment due to torque is then:

MQ =

j

-

and this acts in a plane normal to the propeller axis.

(7.4)

f


-'-dQ

143

-' dT

rZ

Z

I\ \

Figure 7.1: Bending Moments due to Thrust and Torque.

Example 1 A three-bladed propeller of 3.0 m diameter has a thrust of 360 kN and a torque of 300 kN m. Determine the bending moments due to thrust and torque in the root section at 0.3 m radius, assuming that the thrust and torque are uniformly distributed between this radius and the propeller blade tip.

!

.~."..- --- ,


144 Z

=3

ro

= 0.3m

D

= 3.0m

= 360kN

T

Q = 300kNm

dT dQ ([;: and ~ constant.

Hence:

T = dT dr

=

l

R

ro

dT dT - d r = -(R-ro) dr dr

T R-ro.

=

360 -1 1.50-0.30 = 300kNm

Similarly, dQ Q 300 = 250kN = = 1.50 - 0.30 dr R-ro MT =

l

R

1 dT

-Z -d (r - ro) dr =

ro

r

0.30

= 100 ( 0.5 r 2-. 0.3 r ) 11.50 0.30 MQ

=

l

R

ro

250 "'"3

dQ -r1 (r Z dr

11.50

ro) dr

1

- 300 (r - 0.30) dr 3

= 72.000 kN m

1 (0 30) = 11.50 X 250 1- - ' 0.30 3 r

dr

1.50

(r -0.3 In r)1 0 .30 = 59.764kNm

It is often convenient to express the bending moments due to thrust and torque in terms of non-dimensional coefficients. Putting: x =

l'

R

(7.5)

in Eqns. (7.2) and (7.4), one obtains:

(7.6)


145

and lIfQ =

If TO =

Xo

pn2

D511.0 dKQ

Z

xo

x - xo - - - - - dx dx x

(7.7)

R is the radius of the root section, then:

1 1

KT =

Xo

dK -_Q dx :to dx

1

KQ =.

dKT dx dx

.

1

(7.8)

so that:

1 1

]{Tpn

2 D5

2Z

.

xo

KT _dd_ !(x -

XO)

X

dx

1

--=-:"--:-'"'7'"----

1

dKT dx Xo dx

and

(7.10)

The evaluation of Mr and MQ thus depends upon the distribution ofthrust and torque over the radius. A linear distribution is sometimes assumed. However, circulation theory calculations indicate that in most propellers the thrust and torque distributions may be approximately represented by:

(7.11) (7.12) where k 1 and k 2 are constants. Substituting these expressions into Eqns. (7.9) and (7.10), the bending moments due to thrust and torque be come:

t:ol

,

_

146


=

KTpn 2 D5 16 - 6xfi -10xg

6Z

8 + 12 Xc

+ 15 x5

8 - 2 Xc 8 + 12 Xc

6 x5 + 15 x5

(7.13)

and

H

= K Q p n 2 D5

Z

- Q

-

(7.14)

In many propellers, the root section may be assumed to be at 0.2R, so that for such propellers XQ = 0.2 and: 2

AfT -

0.2376 KT pn D Z

MQ - 0.6691

K'

Q P;

5

TD

= 0.2376 Z

2D 5

Q

= 0.6691 Z

(7 l15) (7.16)

~.

it ~

~.

~ iii.'

~

Example 2 A three-bladed propeller of diameter 3.0 m has a thrust of 360 kN and a torque of 300 kN m at 180 rpm. The thrust and the torque may be assumed to be linearly distributed:

=

between! the root section at x 0.2 and x = 1.0. Determine the bending moments due to thrust and torque at the root section. How do these values compare with the values obtaineq by using the distributions of Eqns. (7.11) and (7.12)?

Z = 3

D = 3.0m

T = 360kNm

n = 180 rpm = 3.05- 1

Q = 300kN dK Q dx ... l{T

=

xo = 0.2

360 T = pn 2 D4 = 1.025 x 3.0 2 x 3.0 4 = 0,4818

.. )

Strength of Propellers K

-

Q -

KT =

147 300

=

Q

pn2 D5

----::-----=-. 1.025 X 3.0 2 X 3.0"

fl.O ddKT dx =

Jxo

x

= 0.1338

{l.0 k 1 X dx = 0.4818

JO•2

.

•

that is:

k1

X

2 1 0

/ .

2 0.2

= '12 k1 (1.0 ~ 0.04) = 0,48 k = 0.4818 1 1.00375

!

KQ =

1l.

0dK -d Q dx

,

= 11.0 k 2 x di = 0.2 '

x

Xo

0,48k2

= 0.1338

k2 = 0.27875

MT =

=

\

pn 2 D5 2Z 1.025

l. 1 "'0

=

pn 2 D5 Z

= 1.025

X

_

3.0

X

(1-x

3

5

11.0 0,2

-O.1x 2

3

l1.

O

3.0 3

(

1.00375 x (x _ 0.2) dx

r'

O

=

dx

2

x

0

5

X

3.0 1l. 0.27875 x x - 0.2 dx 0.2

~2 - 0.2 x 2

88.003kNm

0.2

dKQ x - Xo -~--dx

"'0

= 208.2890

J..

dKT - - (x - xo) dx dx

2

3.0 2X 3

X

= 375.0135 MQ

O

r'

X

O

0.2

=

66.652kNm


148

Using the thrust and torque distributions of Eqns. (7.11) and (7.12) with XQ ,\IT

=

0.2376 T:

= 0.2, ,

= 0.2376 3603x 3 = 85.536 kN m

.UQ = 0.6691 Q = 0.6691 300 = 66.910kNm Z 3 (Compare these results with those of Example 1 in which uniform thrust and torque distributions have been used.)

7.3

Bending Moments due to Centrifugal Force

In addition to the bending mom,ents due to thrust and torque, bending moments in planes parallel to the'propeller axis and normal to it also arise due to the centrifugal force on each blade. If a is the area of the blade section at radius r. the mass of the propeller blade between a radius ro and the blade tip is given by:

(7.17)

where Pm is the density of the propeller material. The centroid of the pro pell~r blade will be at a radius:

," ~ 1: l

or dr

(7.18)

R

a dr

ro

so that the centrifugal force on the blade will be:

Fe

=

mb f (21T n)2

=

(21T n)2 Pm

l

R

a f' dr

(7.19)

ro

If the distances between the centroid C of the blade and the centroid ,Co of the blade section at radius ro are measured as shown in Fig. 7.2, the bending

,

.

:

\

151


0.7 0.8 0.9 1.0

l l

R

a dr

ro

-48M-

a 0.0538 0.0358 0.0168 0

-x -

= -13 X

R

2 4 1

1.50 10

-

X

1.3695

f(mb) r 0.15064 0.05728 0.06048 0

f(mb) 0.2152 0.0716 0.0672 0 1.3695

0.68942

= 0.068475 m 3

1.50 31 X 10 x 0.68942 x 1.50

ar dr =

= 0.0517065m4

ro

l l

R ar dr

r=

o

Radr

=

0.0517065 . 0.068475

=

0.755 m

ro

mb = Pm

l

R a dr = 8300 x 0.068475 = &68.34 kg

ro

Fe = mb r (27rn)2 = 568.34 x 0.755 x (21T x 3)2 kgms- 2

= 152.461 kN \

M R = Fcz e = 152.461 x 0.150 = 22.869kNm 'j

Ms

7.4

=

Fe Ye

=

152.461 x 0.035

=

5.336 kN m

Stresses in a Blade Section

The bending moments on the blade section at radius TO due to thrust ami torque and those due to centrifugal force, illustrated in Figs. 7.1 and 7.2, are shown in Fig. 7.3 with reference to the blade section and its principal axes (xo- and Yo- axes). The components of the resultant bending moment along the principal axes are then: '

1

_


152 AXIS PARALLEL TO PROPELLER AXIS - ...

I

I I

I I

/ CLOCKWISE MOMENTS ABOUT AN AXIS ARE POSITIVE

Figure 7.3: Bending Moments at a Blade Section.

M xo M yo

=

-(MT + MR) cos ep - MQ sinep

(7.22)

(MT+MR) sinep-MQ cosep

(7.23)

in which ep is the pitch angle of the blade section,. and the bending moment due to skew has been neglected.

If I xo and Iyo are the moments of inertia (second moments of area) of the blade section about the· Xo- and Yo- axes, and ao the area of the section, one may determine the stress due to the bending moment and the direct tensile stress due to the centrifugal force at any point of the section whose coordinates are (xo, Yo):

;-, , ,:' ·1:.··.··.···'.·

~....'"~ ·'1


149

I

Centroid , . of Blode Centroid of. Section

"0

at radius

Figure 7.2: Bending Moments due to Certifugal Force.

.

.

moments due to the centrifugal force in planes through the propeller axis and normal to it are respectively:

(7.20)

Ms -

Fc'yc

(7.21)

The bending moment MR arises due to the rake of the propeller blades and acts in the same direction as the bending moment due to the propeller thrust MT in propellers with blades raked aft. If the blades were raked forward so that the line of action ofthe centrifugal force passed through the centroid of the section at radius TO, i.e. if Zc = 0, the bending moment due to centrifugal force in a plane through the propeller axis would be zero. The . bending moment Ms arises from the skew of the propeller blades and acts in a direction opposite to the bending moment due to the torque MQ in propellers with skewed back blades. In propellers with' moderate skew, the bending


150

moment due to skew is small and may be neglected, particularly since the error due to this overestimates the resulting bending moments and yields conservative stress values. Moreover, the existence of a bending moment due to skew contradicts the assumption made earlier that the distribution of loading across the blade is ignored. In propellers with heavily skewed blades such an assumption is obviously' untenable.

I

The areas of blade sections at various radii of a propeller of 3.0 m diameter are as follows: 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0.0651

0.0802

0.0843

0.0807

0.0691

0.0538

0.035B

0.0168

0

The propeller runs at 180 rpm. The propeller is made of Manganese Bronze with a density of 8300 kg per m 3 . Determine the centrifugal force on the blade if the root section is at O. 2R. If the centroid of the section is at distances of 0.150 m and 0.035 m from the line of action of the centrifugal force measured parallel and perpendicular to the propeller axis, determine the bending moments due to rake and skew.

\

D

=- 3.0m

:to

=- 0.2

Pm = 8300kgm- 3

n = 180 rpm = 3.0 s-1 Zc

=- 0.150m

Yc = 0.035m

f

mb = jR pm a dr

=

loR ar dr

j

ro

R

a dr

ro

mb and

f

j

I

Example 3

r/R

i

are calculated using Simpson's Rule as follows: x

a

0.2 0.3 0.4 0.5 0.6

0.0651 0.0802 0.0843 0.0807 0.0691

8M -1 4 2 4

2

!(mb)

!(mb) f

0.0651 0.3208 0.1686 0.3228 0,1382

0.01302 0.09624 0.06744 0.16140 0.08292

~


153

(7.24)

>i>1

a positive stress indicating tension and a negative stress indicating compres sion. It is usual to calculate the stresses at the leading and trailing edges and at the face and back at the position of maximum thiCkness of the blade section. For sections of normal aerofoil shape the maximum tensile and compressive stresses occur at the face and the back respectively, close to the position of maximum thickness of the section. The maximum tensile stress in the' root section due to bending is then equal to the bending moment M xo divided by the section modulus Ixo/yo where Yo is the distance of the centroid from the face chord.

, l

,

, I

Example 4 A' propeller of 3.0 m diameter and constant faceI pitch ratio 1.0 runs at 180 rpm. The bending moments due to thrust and torque are respectively 65.700kNm and 59.800 kN m. The mass of each blade is 570 kg, the centroid being at a radius of 0.755m. The centroid ofthe root section at 0.2R is 0.150m forward ofthe centroid of the blade and 0.035 m towards the leading edge from it. The root section has a chord of O.S'OO m, a thickness of 0.160 m and an area of 0.0900 m 2 • The position of maximum thickness is 0.270 m from the leading edge. The centroid of the section is 0.065 m from the face and 0.290 m from the leading edge. The leading and trailing edges at the root section have offsets of 0.020 m and 0.010 m from the face chord. The moments of inertia of the section about axes through its centroid and parallel and perpendicular to the face chord are respectively 1.5 X 10- 4 m 4 and 3.2 x 10- 3 mol. Determine the stresses at the leading and trailing edges, and at the face and the back.

D

MT Zc

I:r.a

=

3.0m

= 65.700kNm = 0.150m = 1.5 X 10- 4 m 4

P

D

MQ

3.0 S-1

1.0

:=

59.800kNm

ffib

0.035m

aa = 0.0900m 2

Yc :=

Iya

n == 180 rpm

:=

:=

3.2

X

10- 3 m 4

:=

:=

570kg

if

= 0.755m


154 Coordinates with respect to the given axes: Face:

:Co

Back:

:Co

Leading Edge:

:Co

Trailing Edge:

:Co

Yo

= 0.020 m Yo = -0.065m

= 0.020m Yo = 0.160 - 0.065 = 0.095 m

= 0.290m Yo = 0.020 - 0.065 = -0.045m

= -0.800 + 0.290 = -0.510 m

= 0.010 - 0.065 = -0.055m = 0.290 - 0.270

= 152.906kN

MR

= Fcze

Ms

= Fc Ye = 152.906 X 0.035 = 5.352 kN m

= 152.906

tanip = P/D =. 1.000 7i:C 11" X 0.2

X

0.150

= 22.936kNm

= 1.5915

Mz;o = -(MT + MR) cosip - (MQ

. ip = 57.858deg

-

Ms) sinip

= -(65.700 + 22.936) 0.5320 - (59.800 - 5.352) 0.8467

= -93.255kNm MilO = (MT

+ M R ) sin

(MQ - Ms)

= (65.700 + 22.936) 0.8467 -

cos
(59.800 - 5.352) 0.5320

= 46.082kNm 1 Stress:

1 j


155

At the leading edge:

s-

1.5

X

-93.255 _ 46.082 10- 4 /(-0.045) 3.2 x 10-3/0.290

= 2.5499 x 10 4 kN m -2

:;:

+

152.906 kN -2 0.0900 m

25.499 N rom- 2

Similarly, stresses at the other points are obtained as: Trailing edge

S::= 43.236 N mm- 2

Face

S = 41.821Nmm- 2

Back

S

7.5

::=

-57.651Nmm- 2

Approximate Methods

Owing to the comparative complexity of the method to determine propeller blade stresses discussed in the preceding sections, various approximate meth ods have been proposed. Such methods have been found to give satisfactory results and are sometimes used in the preliminary stages of propeller design . when all the design details are not known. Two such methods are considered here. A widely used approximate method is due to Admiral D. W. Taylor (1933), who considered .the problem of propeller blade strength in great· detail but by making various assumptions succeeded in reducing the problem to a few formulas for estimating the maximum compressive and tensile stresses in the root section of the propeller blade. The major assumptions in Taylor's method in addition to those given in Sec. 7.1 are: (i) The thrust distribution along the propeller radius is linear. (ii) The ma.ximum thickness of the blade also varies linearly with radius. (iii) The root section is at O.2R. (iv) The propeller efficiency is a linear function of the apparent slip in the normal operating condition.

J..----


156

Based on these assumptions, the maximum compressive and tensile stresses in the root section due to thrust and torque are given by formulas, which can be put into the following form: COPD

Se =

(7.25)

2

ZnD3; (;) ST = Se (0.666 + C1

~)

(7.26) .

The additional compressive and tensile stresses due to centrifugal force are given by: / S c' . -- C 2Pm n 2 D· 2

[C3 taUE: _ .] t 1 2Y. D

BT'

=

C 2Pm n 2

D2lrC3 taUE: t + CemtanE: + .1] 4

3Y.

D

~

D

(7.28)

. I

where:

!

Co, Cl, C2, C3, C4

=

coefficients dependent on the pitch ratio P/ D

PD

=

delivered power

n

=

revolution rate

z

number of blades

D

diameter

c

D

to D

=

chord-diameter ratio of the root section

=

blade thickness fraction

i t

I

1

I i

Strength of Propellers t

c

Pm c Cmax

D

157

-

thickness-chord ratio of the root section

=

density of propeller material

=

rake angle

=

maximum chDrd-diameter ratio of the propeller.

Equations (7.25)-(7.28) are dimensionally ,homogeneous. However, the values of the coefficients in Table 7.1 give the stresses in kN per m 2 if PD is in kW, n in revolutions per sec, D in m and Pm in kg per m 3 • Table 7.1 Coefficients for Taylor's Method P

Co

C1

C2

C3

C4

0.600

7.499

0.650

0.002568

2.750

1.590

0.700

6.471

0.710

0.002568

2.600

1.690.

0.800

5.659

0.754

0.002568

2.400

1.790

0.900

5.073

0.784

0.002568

2.200

1.870

1.000

4.583

0.804

0.002568

2.070

1.925

1.100

4.190

0.817

0.002568

1.920

1.980

1.200

3.895

0.823

0.002568

1.800

2.020

1.300

3.674

0.820

0.002568

1.690

2.050

D

Taylor's method has been found to give satisfactory results for propellers with normal blade outlines and moderate blade area ratios. For very large blade area ratios the method gives stress values which are 10-15 percent lower than values obtained by more accurate methods. Another approximate method for estimating propeller blade stress is due to Burrill (1959). In this method it is assumed that the thrust distribution

...._-------

·, Basic Ship 'Propulsion

·158

is such that the thrust on each blade can be taken to act at a point whose distance from the root section is 0.6 times the length of the blade from root to tip. The transverse force on each blade which gives rise to the torque is similarly taken to act at a distance from the root section of 0.55 times the length of the blade. The thrust and torque bending moments can therefore be written as:

MT

-

T TD Z x 0.60 (R - ro) := Z x 0.30 ( 1 - Xo )

Q x 0.55 (R -

TO )

MQ - Z[0.55 (R - ro) + ro] where ro

=

XQ

=

Q x 0.55 (J - Xo ) Z (0.55 + 0.45 Xo )

R is the radius of the root section.

:'

(7.29) , (7~30)

.

The mass of each blade is approximated by:

mb

-AD = Pmt-zk

,

(7.31)

where f is the mean thickness of the "blade from root to tip, AD the developed blade area and k a coefficient. For a linear distribution of thickness, (7.32) where tolD is the blade thickness fraction of the propeller and tl is the blade thickness at the tip. A value of k = 0.75 is often used. The distance of the centroid of the blade from the root section is taken as 0.32 times the length of the blade from root to tip for blades with normal outlines and 0.38 times the blade length for blades with wide tips, i.e. (7.33)

where f = x R is the radius of the blade centroid, k1 = 0.32 for normal blade outlines and kl = 0.38 for wide tipped outlines.


159

The centrifugal force is then given by:

(7.34) The bending moment due to rake is:

MR

=

Fe (x -:- xo) R tancE

(7.35)

where cE is the effective rake angle, about 6 degrees greater than the geo metric rake angle. The effect of skew is neglected. The cross-sectional area of the root section and its section modulus are estimated as follows:

(7.36) (7.37)

where c and t are the chord and thickness of the root section, and k2 and k3 are coefficients whose values are as follows: . Section Shape Segmental

0.667

0.112

Aerofoil

0.725

0.100

Lenticular

0.667

0.083

\

The stress in the root section is then given by:

S = (MT+MR) cos
+ Fe a

(7.38)

y

where

1__ . ~

160


Example 5

(a) A four-bladed propeller of 5.0 m diameter has a constant pitch ratio of 0.950, an expanded blade area ratio of 0.550 and a blade thickness fraction of 0.045. The root section is at 0.2R and has a chord-diameter ratio of 0.229 and a thickness-chord ratio of 0.160. The maximum chord-diameter ratio of the blade is 0.301. The propeller blades have a rake of 10 degrees aft. The pro peller is made of Nickel Aluminium Bronze, which has a density of 7600 kg per m 3 • The propeller has a delivered power of 5000 kW at 120 rpm. Deter . mine the propeller blade stresses by Taylor's method.

(b) Determine the propeller blade stress by Burrill's method given the following additional data: speed of advance 7.0 m per sec, propeller efficiency 0!690 and blade thickness at tip 17.5mm. The propeller blades have a normal outline and aerofoil sections. ! (a) Taylor's method:

Z

=4

D

Xo = 0.2 €

= C

D

5.0m

P D

-t =

= 0.229

=

= 10" Pm

= 0.950 0.160

C

7600kgm- 3

AE to = 0.550 D Ao Cmax

D

::::

::::

0.045

0.301

FD:::: 5000kW

n = 120rpm :::: 2.0s- 1

P From Table 7;1 for D

= 0.950,

Co = 4.828 C3 = 2.135

Se =

C1 = 0.794 C4 = 1.898

COPD

ZnD3

~ (~r

= 52057kNm- 2

=

:=

C 2 :::: 0.002568

4.828 x 5000 -4-x-2-.0-x-5.0 3 x 0.229 X (0.045)2

52.057Nmm- 2


= Se

ST

161

( 0.666 + C 1

~ ) = 52057 ( 0.666 + 0.794 x 0.160 )

2 = 41283kNm- = 41.283Nmm- 2

3 tanc ] Se = C 2Pm n 2 D2 [C2tofD-1

[2.135 tan 10° _ 1 ] 2 x 0.045 .

= 0.002568 x 7600 x 2.0 2 x 5.02

= 6212 kN m -2 = 6.212 N mm- 2 S'

=

T

C 2Pm

3 n 2 D2 [C tane 3to/D

+

= 1951.68 [2.135 tan 10° 3 x 0.045

= 9564kNm- 2

C4 tane cmax/D

Tensile stress,

+ Se ST + Sir

Se

]

+ 1.898 tan 10° + 1] 0.301

= 9.564Nmm- 2

Compressive stress,

+1

;'

= 58.269 N rom -2. = 50.847Nmm- 2

(b) Burrill's method: \

'17

= 0.690

tl :::: 17.5mm

Normal blade outline, aerofoil sections

k

= 0.75

k2

k1 = 0.32

::::

0.725

k3

::::

PT = Pn'17 :::: 5000 x 0.690 :::: 3450 kW T::::

Q

._-------~-

::::

PT

::::

VA

3450 7.0

5000 :::: 397.887 kN m X 2.0

Pn

-- = 27rn 27r

---

= 492.857kN

'-'

0.100

:"1' I

162


.. ~

Mr = T: x 0.30 (1- xo) = _49_2_.8_S;:--X_S_.O x 0.30 (1 - 0.20) = 147.857kNm M

Q

= Q x 0.55 ( 1 - Xo ) Z 0.55 + 0.45 Xo

=

397.887 0.55 (1 - 0.20) x . 4 0.55 + 0.45 x 0;20

= 68.387kNm

tl

17.5

D = 5000

t

= 0.50

= 0.0035

[( 1 - xo) ;

4- (1 + xo) i]

D

= 0.50 [( 1 - 0.20) 0,45 + ( 1 + 0.20) 0.0035] 5.000 m

= 0.1005m

AE

=

mb ;:;

AE

Ao

11'

D2 = 0.55 x 4

11'

2 x 5.0 4

=

10.799m2 ~ AD

-AD pmt Z k

10.799

= 7600 x 0.1005 x - 4 - x 0.75 ;:; 1546.584kg

x R =

[xo

+ kI( 1 -

Xo )] R = [0.20 + 0.32 ( f - 0.20 )J 2.500

;:; 1.140m

Fc =

mb

(211'n)2 fiR ;:; 1546.584 x (211' x 2.0)2 x 1.140N

;:; 278418.5 N = 278.419 kN

, .,)1' "~

J"

;i


=

MR

163

Fe ( x -

XQ )

R tan c E

278.419 (0.456 - 0.20) x 2.500 tan ( 10 + 6)° kN m 51.095 kN m

a

=

= k2 (~r~D2 =

k 2 et

= 0.1521 m

~ =k =

3

ct 2

3.8429

=

Stress

7.6

i I, i

=

k3

0.229 2 xO.160

X

5.02

(~) 3 G) 2 D = 0.100 X 0.229 3 x 0.1602 X 5.0 3

X 10-3

3

m3

56.520°

s =

X

2

tan ep

ep

0.725

(AfT

=

PID 11" XQ

cos ep

=

0.950 11" x 0.20

=

0.551'6

=

1.5120

sin ep

=

0.8341

+ MR) cosep + MQ sinep + Fe fly

a

=

( 147.857 + 51.095 ) 0.5516 + 68.387 x 0.8341 3.8429 x 10- 3

=

45230kNm- 2 = 45.230Nmm- 2

278.419

+ 0.1521

Classification Society Requirements

Classification Societies such as the American Bureau of Shipping and the Lloyd's Register of Shipping prescribe the strength requirements that pro pellers must fulfil. These include requirements for the minimum blade thick ness, the fitting of the propeller to the shaft, and the mechanical properties of the propeller material. Lloyd's Register (LR), for example, specifies the minimum propeller blade thickness at O.25R and a.GOR for solid propellers (i.e. propellers in which the


164

blades are cast integral with the boss, unlike controllable pitch propellers). For a propeller haYing a skew angle less than 25 degrees, the blade thickness neglecting any increase due to fillets is given by a formula which, in LR's notation, is:

T

=

KG A

EFU EN

+ 100

rl-l50M

P

EFRU LN

(7.39)

where:

K

GBD 3 R 2 675

gl cm 3

G

-

density of the propeller material in

B

-

developed blade area ratio

D

-

propeller diameter in m

R

propeller rpm at maximum power

G

-

A

-

rake at blade tip in mm (positive aft)

E

-

actual face modulus/0.09T2L, but may be taken as 1.0 and 1.25

respectively for aerofoil sections with and without trailing edge

washback

T

-

blade thickness in mm at the radius considered, i.e. O.25R or 0.60R

1.0 for 0.25R and 1.6 for O.60R

L

length in mm of the expanded cylindrical section at the radius considered

U

allowable stress in N per mm2

F

.:.V

=

PO.25 D + 0.8 for 0.25R

-

PO.6 D + 4.5

-

number of blades

for O.6R

.t

Strength of Propellers M

P

165

-

3.75 1.0 + oR jD

-

1.35 + oR

-

maximum shaft power ir: kW

0.7

5 jD

0.7

PO.25

+ 2.8 D

P

for 0.25R .

+ 1.35 DO 6

for 0.60R

(Note the units of the different quantities.) The fillet radius between the root section and the propeller boss is not to be less than the thickness of the lOot section. Composite radiused fillets or elliptical fillets which provide a greater effective radius to the blade are preferred., Where fillet radii of the required size cannot· be provided, the allowable 'stress must be appropriately reduced. The allowable stress for propellers of cast iron, carbon steels or low alloy steels may be increased if an approved method of cathodic protection is provided. The allowable stress given in Table 7.4 may be increased by 10 percent for twin screws. For propellers with skew angles exceeding 25 degrees, a detailed blade stress computation based on calculated hydrodynamic pressure distributions along the length and width of the blades must be carried out. For propellers operating in more than one operating regime, e.g. tug and trawler propellers, detailed stress computations for each condition must be carried out. There are special requirements for propellers in ships operating in ice covered waters. Detailed requirements for the fitting of the propeller to the shaft are also given. These requirements are important because the propeller shaft and the boss are regions of high stress, which can lead to the loss of the propeller at sea. Special care is required to see that the keys and keyways do not lead to high stress concentrations. Round ended or sled runner keys are to be used, and the corners of the keyways must be provided with smooth fillets of radii at least equal to 0.0125 times the shaft diameter. Effective means must be provided to prevent sea water coming into contact with the steel propeller shaft inside the propeller boss, since this reduces the fatigue strength of the shaft significantly.

i

<

I ;,~

/'

,t ii


166

rt<.

, -;~i~}

'-f!.";,

7.7

Propeller Materials

'~: ~;~~

i

__

The material of which a propeller is made should have certain desirable properties. The processes involved in the manufacture of a propeller are casting and machining, and the propeller material must be amenable to these processes. It may be necessary to consult the propeller manufacturer when deciding on a propeller material. The material should be easy to cast into large castings, which should have uniform shrinkage and low internal stresses. The propeller casting should be capable of being machined to a high degree of accuracy and given a smooth surface with a high degree of polish. The propeller material should have a high strength and toughfJ.ess so that the propeller blade thickness is not so large as to impair efficiency. Fatigue strength is also importa,nt. Resistance to corrosion in sea irater and to erosion are desirable qualities in a propeller material. It i~ also advantageous if the material is such that a damaged propeller can be easily repaired. A low density allows a propeller to be made lighter. The' cost of the propeller material is also a consideration. The first material to be widely used for ship propellers was cast iron, which is easy to cast and has a low cost. However, it has a low strength and ductility, a low resistance to corrosion and erosion, cannot be finished to a smooth surface and cannot be easily repaired. Cast iron propellers have thicker blades and are less efficient. Spheroidal graphite cast iron, which is stronger and more ductile, is preferred. However, for propellers in oceangoing ships, copper alloys have largely superseded cast iron as a propeller material, except in some tugs, ice breakers and similar vessels in which cast iron is used for propellers because the blades break off cleanly when they strike an obstacle without damaging the rest of the propulsion system. Cast steel is also used sometimes for making propellers. However, cast steel propellers have a fairly rough surface finish and run the risk of corrosion and have reduced fatigue strength in sea water. Copper alloys ranging from l\'langanese Bronze to Nickel Aluminium Bronze are now widely used as propeller materials since they have the desired qualities. There is some risk of dezincification with Manganese Bronze. Nickel Aluminium Bronze has a high fatigue strength. Stainless steels are somewhat difficult to cast but are highly resistant to corrosion and erosion and have high strength and toughness.

, '

1

j


167

Classification Societies generally· specify the chemical composition and me chanical properties of the materials which may be used for propeller man ufacture. Lloyd's Register, for instance, specifies the chemical composition for carbon-manganese steels for cast steel propellers given in Table 7.2. Table 7.2 Chemical Composition of Carbon-Manganese Steels for Cast Steel Propellers (Lloyd's Register of Shipping)

Carbon

0.25% max

Residual elements:

Silicon

0.60% max

Copper

0.30% max

Manganese

0.50-1.60%

Chromium

0.30% max

Sulphur

0.040% max

Nickel

0.40% max

Phosphorus

0.040% max

Molybdenum

0.15% max

Total

0.80% max

For alloy and stainless steels, the chemical composition, heat treatment, mechanical properties, microstructure and repair procedures must be sub mitted for approval. The chemical composition of copper alloys used for propellers is given in Table 7.3, while the mechanical properties of materials normally used for propeller manufacture are given in Table 7.4.

\

7.8

Some Additional Considerations

,.

Although the propeller blade thickness is determined primarily from con siderations of allowable stress, propeller designers often adopt higher blade thicknesses than those determined by strength calculations. Apart from providing a greater margin against structural failure, a higher blade thick ness also provides a greater margin against cavitation. This is evident from Fig. 6.7.

I!

In heavily loaded propellers operating in a non-uniform wake, it is also necessary to consider fatigue strength. The minimum and maximum values

I-' Q')

,Table 7.3

00

Chemical.Composition of Copper Alloys for Propellers

(Lloyd's Register of Shipping) Chemical Composition (percent)

Alloy Designation Grade Cu 1 Manganese Bronze (High Tensile Brass) Grade Cu 2 Nickel Manganese Bronze (High Tensile Brass) Grade Cu 3 Nickel Aluminium Bronze Grade Cu 4 Manganese Aluminium Bronze

Cu

Sn

Zn

Ph

Ni

Fe

Al

Mn

52-62

0.1-1.5

35-40

0.5 max

1.0 max

0.5-2.5

0.5-3.0

0.5-4.0

50-57

0.1-1.5

33-38

0.5 max

2.5-8.0

0.5-2.5

0.5-2.0

1.0-4.0

77-82

0.1 max

1.0 max

0.03 max

3.0-6.0

2.0-6.0

7.0-11.0

0.5-4.0 tx:l

~. o 70-80

1.0 max

6.0 max

0.05 max

1.5-3.0

2.0-5.0

6.5-9.0

8.0-20.0

~ '0"

1~ g.

l~ '.,

.

---

•

-

-

In

~'Rd'V?2M9"sfF%W

sssm __,,,'_::V,",~

~~

tr.l

Table 7.4

I

Mechanical Properties of Propeller Materials (Lloyd's Register of Shipping)

Material

Grey Cast Iron Spheroidal or Nodular Graphite Cast Iron Carbon Steels Low Alloy Steels 13% Chromium Stainless Steels Chromium-Nickel Austenitic Stainless Steel Duplex Stainless Steels Grade Cu 1 Manganese Bronze (High Tensile Brass) Grade eu 2 Nickel Manganese Bronze (High 'Tensile Brass) Grade Cu 3 Nickel Alu..TIlinium Bronze Grad.e Cu 4 Manganese Aluminium Bronze

g,

Minimum Tensile Strength

Density

N/mm2

g/cm3

N/m.m2

250 400 400 440 540 450 590 440 440 590 630

7.2 7.3 7.9 7.9 7.7 7.9 7.8

17.2 20.6 20.6 20.6 41.0 41.0 41.0 39.0 39.0 56.0 46.0

Allowable Stress

::p

.g (ll

~

8.3 8.3

7.6

7.5

~

....

C)

co

170

Basic Ship Propulsion FATIGUE LIMIT fOR B

tO

CYCLES

ALTERNAnNG

STRESS

O'---.L.--'---'---'---'----'--'--'--.L.~___'__'___'__J'__'__=r_-'-----'---'-...J

o

rIME AVERAGE STRESS

YIELD STRESS

Figure 7.4 : Allowable Alternative Stress. l

I

of the bending moments due to the thrust and torque of a propeller wo~king in a non-uniform wake field can be determined using lifting line or lifting surface theory, and the minimum and maximum stresses at the blade root calculated. The allowable alternating stress for a given number of cycles depends upon the temporal mean stress) and can be determined by a Good man diagram, Fig. 7.4. The fatigue strength of some propeller materials in sea water is given in Table 7.5 along with their yield stress or 0.2 percent proof stress. For heavily skewed propellers, the stresses calculated by the methods dis cussed earlier, i.e. using the beam theory and neglecting the distribution of loading in a chordwise direction, do not correlate well with experimental data. The beam theory calculations predict neither the magnitude nor the location of the maximum stress at the blade root correctly. This is partly due to the fact that in addition to the bending moments due to thrust and torque there are also twisting moments because the centre of action of the hydrodynamic forces does not lie close to the radial line through the cen troid of the root section. It is thus necessary to consider both bending and torsion in heavily skewed propeller blades. Lifting surface calculations, or lifting line calculations in conjunction with two-dimensional pressure distri-


171 Table 7.5

Fatigue and Yield Strengths of some Propeller Materials Material

Fatigue Strength for 108 cycles in Sea Water

Yield Stress or 0.2% Proof Stress

NJmm 2

Manganese Bronze

41

Nickel Manganese Bronze

41

175 175

Nickel Ahl,minium Bronze

86

245

Manganese Aluminium Bronze

62

275

bution calculations, are used to determine the hydrodynamic loading on the blades of highly skewed propellers, and .then finite ~lement procedures are used to determine the blade stresses. A modified form of the beam theory has al::o been used for estimating the stresses in highly skewed propeller blades. Boswell and Cox (1974). The axis of the beam is assumed to be a radial line through the centroid of the cylindrical blade section at which the stress is being calculated. The blade section shape is modified to a thickness distribution at zero camber since it has been seen that this gives better cor relation with experiments. .The hydrodynamic loading at each radius then produces a force normal to the axis of the beam and a moment (torsion) about the axis. The centrifugal loading, on the other hand, produces a force parallel to the axis and a bending mom~nt perpendicular to the axis. The forces and moments on the root section of a propeller blade with rake and skew are shown in Fig. 7.5. Hea\-ily skewed propeller blades may also suffer from a form of structural instability in which the loading causes the blades to deflect in such a way that the effective pitch is increased resulting in a further increase in loading and so on. A large skew also reduces the natural frequency of the propeller blades and may lead to stresses due to blade vibration.

ro


172

-.1-dQ

rZ

Moments .at Rool Section (radius r o ) due to Forces on Section at radius r

1dT (r-ro)

Bending Moment due to Thrust

dM T =

Bending Marnen t due to Torque

dM Q = rzdQ(r -ro cos1/!E)

Bending Moment due to Centrifugal Force and Roke

dM R

I

=

Bending Mamen l due to Cen trifugal Force and Skew

dFC ro sin 1JrE

Twisling Marne'll due to Thrusl

dN r

Twisting Moment due to Torque

dNa

FE

dIe (r -ro) Ion EE

telTro sin1Jr[

=

ylZ dQ(r -r o) Ion E[

and We are the effective rake and skew angles.

The

centre of hydrodynamic pressure and the centroid of lhe blode section are assumed to be coincident. Fig~lre 7.5

:Forces and Moments in a Blade with Large Ralle alld Shew.

The thickness of a propeller blade at the tip and at the edges should be sufficient to prevent the blade from bending due to the hydrodynamic loading since the resulting change in section shape may cause an increase in the effective pitch of the propeller and lead to cavitation erosion. Experience has shown that the blade thickness at the tip before rounding should be between O.003D and O.004D. The blade edge is normally rounded and has a radius which reduces from between O.OOI75D and O.002D at O.2R to about O.0006D at O.95R. The blade thicknesses at the leading and trailing edges before rounding are given approximately by:


tLE

D tTE

D

173

r

= 0.0058 - 0.0043 R

-

0.0030 - 0.0015 ~

(7.40)

The dimensions of the propeller boss are determined by the geometry of the propeller blades and the diameter of the propeller shaft. The boss diameter is usually between 0.15D and 0.20D, and the length is just sufficient to accommodate the blade at the root. The diameter of the boss at the forward end is 10-15 percent more than in the middle, and the diameter at the after end less than that in the middle by about the same amount. An internal recess may be provided in the boss, in which case the length of this recess should not exceed one-third the length of the boss, and the minimum wall thickness should not be less than the thickness of the blade root section. The blades should join the boss through fillets of adequate radii, usually 0.03 0.04D but not less than the thickness of the root section. A streamlined boss cap or "cone" is usually fitted at the after end of the boss.

Problems 1. A four-bladed propeller of 6.0 m diameter and 0.9 constant pitch ratio has a thrust of 1100 kN and a torque of 1250 kN m at 150 rpm. Each blade of the propeller has a mass of 1800kg with its centroid at a radius of 104m. The thrust and torque distributions in a non-dimensional form can be assumed to be given by: dKQ = dx The root section at 0.2R has a section modulus of 7.5 x 106 mm3 and an area of 2.5 x 105 mm2 • Determine the stress in the root section, neglecting the bending moments due to centrifugal force. 2. In a three-bladed propeller of 4.0 m diameter and 0.7 pitch ratio, the thrust and torque distributions are linear between x = 0.2 and x = 0.5, and constant from x = 0.5 to x = 1.0 as follows:

J-

_

174

Basic Ship Propulsion x = 0.2: x

=

0.5:

dKQ dx

= 0.012

dKQ dx

= 0.030

The propeller runs at 180 rpm. The root section at 0.2R has a thickness chord ratio tic 0.250 and a section modulus of 0.105ee. The thickness of the .propeller blades at the tip is 15 mm. Determine the thickness and chord of the root section and the blade thickness fraction of the propeller if the thickness distribution from root to tip is linear and the stress due to thrust and torque is not to exceed 40 N per mm2.

=

i

3.

A ship has a four-bladed propeller of 5.0m diameter and 0.8 constant pitCh ratio. The propeller develops a thrust of 800kN and a torque of 700 kN m'at 150 rpm. The radial distributions'of thrust and torque are as follows: 1

Each blade has a mass of 1500 kg. The blades are raked aft and skewed back in such a way that the centroid of each blade is at a radius of 1.2 maud at distances of 0040 m and 0.25 m respectively from two reference planes, the first passing through the propeller axis and the other being normal to it. The root section at 0.2R has an area of 2 x 105 mm2 and a minimum section m~dulus of 6.5 x 106 nun3 , its centroid being at distances of 0.15 m and 0.05 m respectively from the two reference planes. Determine the maximum blade stress. 4.. A ship is to have a four-bladed propeller of 5.5 m diameter and 0.75 pitch ratio with all' expanded blade area ratio of 0.500 and a rake of 15 degrees aft. The propeller IS required to absorb a. delivered power of 7500 k\V at 138 rpm· and 8.0m per sec speed of advance, its efficiency being 0.650. The propeller belpngs to a methodical series for which:

to A E 3 Mass of each blade = 0.0625 Pm D Ao D Centroid of blade from the propeller axis

= OA8R

The root section at 0.2R has a thickness chord ratio tic = 0.250 and the following properties: Area of section

= 0.700ct

Ordinate of leading edge from face chord

= 0040 t

Ordinate of trailing edge from face chord = 0.35 t


r' i

177

A, B, 0, On and Os are coefficients as follows:

=

A

l.O+6.0

B = 4300wa N

o On

Os = PO.7

PO.25 W

a D L O.25

f 10

UJ as

(!!:-)2 (D)3 100

= (1 + 1.5 P~25) =

PO.25

D

R0.7 +4.3J) 20

(L O.25 f - B )

10 but not greater than 0.10 UJ LO. 25 ta.25' L O.25 to.25

= =

Pitch at 0.7 radius in m Pitch at 0.25 radius in m = material constant given in Table 7.6 = expanded blade area ratio = propeller diameter in m = width of the blade at 0.25. radius in m = material constant given in Table 7.6 == trloment of inertia of the section at 0.25 radius about an axis through the centre of .area and parallel to the pitch line of the section in mm4 = maximum distance of the section axis from.. the face of the section in mm = area of cross-section at 0.25 radius in mm2

Table 7.6 Material Constants Material Manganese Bronze Nickel Manganese Bronze Nickel Aluminium Bronze Manganese Aluminium Bronze Cast Iron Carbon and Low Alloy Steels

Grade Grade Grade Grade

Cu Cu Cu Cu

Minimum Ultimate Tensile Strength N/mm2 440 1 440 2 3 590 4 630 250 400

f

w

20.6 20.9 25.7 23.25 11.77 14.0

8.3 8.0 7.5 7.5 7.2 7.9

178

Basic Ship Propulsion Determine the blade thickness fraction of the propeller whose particulars are given in Problem 8. Take Cn 0.090 and Cs 0.700.

=

=

10. A large single-screw oceangoing ship has a speed of 25 knots with a delivered power of 22500 kW at 108 rpm, the wake fraction being 0.220, the thrust deduction fraction 0.160 and the propeller efficiency in the behind condition 0.680. The propeller has six blades and a diameter of 7.0 m. The pitch ratio PI D, the chord diiUIleter ratio clD, the thickness chord ratio tic, the effective skew angle 'l/JE(i.e. the angle between radial lines passing through the mid. chords of the root section at 0.2R and a section at any other radius r, and the effective rake angle cE (the angle which a line joining the centroid of the root section and the centroid of the section at radius r makes with a plane normal to the propeller axis) are given in the following table: r I R:

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

' 1.0

PID:

0.675

0.767

0.836

0.900

0.902

0.900

0.874

0.824

0.750

cl D:

0.201

0.238

0.265

0.282

0.288

0.281

0.256

0.203

tic:

0.199'

0.159

0.128

0.102

0.080

0.061

0.047

0.039

'l/J~:

0

-1.25

0

3.75

10.00

18.75

30.00

43.75,' 60.00

.,.0.

o

3.67

7.00

10.00

12.67

15.00

17.00

18.67

<-E'

20.00

The sectional area at each radius is given by 0.7 ct. The propeller is made of Nickel Aluminium Bronze of density 7500 kg per m S • The thrust and torque distributions are given by: Q dK= k 2X '1 (1 -x )0.5

dx

wl~ile the loading along the chord at each radius may be taken as uniformly distributed over the face. Determine the bending moments about the principal a.xes and the torsion about a radial axis through the centroid of the root section. The centroids of all the blade sections may be assumed to be at mid-chord. Calculate also the tensile force on the root section.

CHAPTER

8

Propulsiorl Model Experiments 8.1

Introduction

The performaD"e of a propeller in a ship and the resulting hull-propeller in teraction are usually determined through model experiments. Experiments with full-size ships and propellers are difficult to carry out and purely theo retical approaches are as yet inadequate for predicting the behaviour of ships and propellers fully. The condit.ions under vv".ich model experiments can be used to determine ship performance have been considered in Chapter 4. It is not practicable to fulfil all these conditions exactly, and therefore some empirical corrections are ·necessary in determining ship performance from experiments with models.' . . The experiments that are normally carried out with ship models and model propellers are: (i) resistance experiments, (ii) .open water experiments, (iii) self-propulsion experiments, (iv) wake measurements, .and (v) cavitation experiments.

179

180


Model experiments in waves are not considered here. Resistance exper iments are also not really within the scope of the present work but are discussed briefly because they are essential for analysing self-propulsion ex periments. The resistance, open water and self-propulsion experiments are normally carried out in towing tanks. Cavitation experiments require a cav itation tunnel or a depressurised towing tank. Wake m~asurements may be carried out in a towing tank or in a circulating water channel.

.J

A towing tank is a long narrow tank in which a ship model is towed through water at a steady speed usually by a towing carriage running on rails along the sides of the tan~. Instruments are provided to measure the speed and the resistance of the ship model, and the thrust, torque and revolution rate of a model propeller if fitted. In a circulating water channel, the ship model is stationary and the water is made to move past it at a steady speed. A cavitation tunnel, which is discussed in more detail in Sec. 8.6, is similar except that the pressure in the tunnel water can be varied over a wide range.

8.2

Resistance Experiments

Resistance experiments are carried out with ship models to determine the resistance of the model and thereby of the ship in a given condition. Such experiments are useful in optimising the hull form and for predicting the power requited for propelling a ship at a specified speed. Before discussing resistance experiments, it is necessary to consider the relationship between the resistance of a ship and the resistance of a geometrically similar model.

It can be shown in a manner similar to that described in Sec. 4.3 that the total resistance of a ship can be expressed in the form:

RT

RT

p£2V2

=

f

(V-;;-'LV) hL

(8.1)

\vhere L is the length of the ship, V its speed, g the acceleration due to gravity, and p and II the density and kinematic viscosity of water respectively. This may be written in terms of non-dimensional parameters as follows:

(8.2)

l

Propulsion Model Experiments

181

where:

CT =

Rn

=

Fn

-

S

RT 2 lpSV 2 .

l1L v

V

y'gL

is the total resistance coefficient,

is the Reynolds number, is the Froude number, and is the wetted surface of the ship, proportional to L 2 • I

Following W. Froude, it is usual to divide the total resistance into a vis cous resistance component which is a function o~ the Reynolds number and a residuary resistance component which is a ftinction of the Froude num ber. This is expressed in terms .of. a viscous re~istance coefficierLt Cv and a residuary resistance coefficientCR: ;

CT = Cv(Rn)+CR(Fn

)

(8.3)

I

i

The viscous resistance coefficient is determined witJl the help of a "fric tion line", Le. an equation giving the resistance of a plane surface in terms of the Reynolds number. Several friction lines have been proposed, based on theoretical considerations and experimental measurements, but the stan dard formulation in use at present is the ITTC (International Towing Tank Conference) 1957 friction line:

CF = 0.075 (loglO

Rn .- 2 )-2

(8.4)

where C F is the frictional resistance coefficient. (The ITTC friction line is not strictly a "two-dimensional" friction line.) The viscous resistance coefficient and the frictional resistance coefficient are related by a "form factor" (1 + k), which accounts for the increase in viscous resistance due to the three-dimensional hull form of a ship:

Cv = (1 + k)CF

(8.5)

One therefore has the following expression for , the resistance of a ship:

182


It is then assumed that the residuary resistance, which for most ships is mainly wave making resistance, obeys the Froude law: the residuary resis tances of geometrically similar ships are proportional to their displacements if their speeds are proportional to the square roots of their lengths. This implies that: . if (8.7)

where the suffix S refers to the ship and M to the geometrically similar model, and F n is the Froudenumber. This allows the total resistance of a ship at a given speed to be estimated from the total resistance of a model at the "corresponding" speed (the speed for which the Froude numbers of the model' and the ship are equal), as illustrated by the following example. Example 1 \

A ship of length 100 m and wetted surface of 2700 m 2 has a speed of 15 knots. A 4.0 m long geometrically similar model of the ship has a total resistance of 25.0 N at the corresponding speed. Determine the total resistance of the ship assuming a form factor of 1.05.

Ls -- 100

LM

= 4.0 m

ill

8s

8M

=

= 8s

v:"'[ ;: :- VS JLM Ls R TM

2700 m 2

=

(

Vs = 15 k

L: )2 =

L

= 7.716

25.0 N

2700 x

= 7.716 ms- 1

(40)2 1~0 =

J 100 4 = 1.5432 ms1+k

25.0 ~ x 1000 x 4.32

X

=

1

1.05

1.5432 2

4.860

X

10- 3

,

,~.


183

·r'

I

!

=

RnM = GFM

G RM =

1.5432 x 4 = 5.419 1.139 X 10- 6

= 0.075 (loglO RnM -

GRS

106

= 3.347 X 10- 3 .

(1 + k) GFM = [4.860 - 1.05 x 3.347] 10- 3

GT M -

= 1.346

2)-2

X

X

10- 3

=

GRM

since

7.716 x 100 VsLs = 1.188 X 10- 6 - 6.495 Vs

X

108

GFS = 0.075 (loglO Rns - 2 )-2 = 1.616

X

10- 3

!

G TS RTS

= =

(1 + k)GFS

+ G RS = [1.05 x 1.616 + 1.346] x 10- 3 = 3.043 X 3 2 GTS ~PS Ss V§ = 3.043 X 10- x '~ x 1.025 x 2700 x 7.716

-

250.7 leN

10- 3

The total resistance RT or the total resistance coefficient CT determined in this manner must be corrected for the differences between the conditions of the model experiment and the operating conditions of a ship. A simple way to carry out this correction is to add a "correlation allowance" CA to the total resista.nce coefficient of the ship. The correlation allowance, which primarily accounts for the roughness of the ship 'surface, is determined by correlating the performance of ships on speed trials and the results of the corresponding model experiments. Other differences between the operating conditions of the ship and the controlled conditions of the model experiments are taken into account by adding a "service allowance" or a "trial allowance" to the resistance of a ship determined through model experiments. More detailed methods may also be used to correct the resistance for dif ferences between the model and the ship. Three such corrections are used in the standard ITTC ship' performance preqiction method (ITTC 1978):

tl----


184

\

(i) The wetted surface of the ship hull is normally rough whereas the

model surface is smooth. It is therefore necessary to add to the viscous

resistance coefficient of the ship a "roughness allowance" given by:

(8.8) where ks is the average amplitude of roughness of the wetted surface of the ship. A standard value of ks = 150 X 10-6 m is used in the ITTC method.

(ii) It is usually impracticable to reproduce bilge keels on a ship model, and therefore a correction is required. This correction is made by increasing the wetted surface,Ss of the ship hull by the surface area SBK of the bilge keels in calculating the viscous resistance of the ship.

(iii) It is also necessary to account for the resistance of the above-water part of the ship. This "air resistance" can be determined by model tests in I a wind tunnel. For a ship moving in still air, the ITTC performance prediction method uses an air resistance coefficient as follows: CAA =

1 RAA 2PSSSVg

=

0.001 AT Ss

(8.9)

. where RAA is the air resistance, CAA the air resistance coefficient and AT the transverse projected area of the above water part of the ship. After making these three corrections, the total resistance coefficient of the ship is given by: Crs =

S8

+ SBK Ss

[(l+k) CFS+f:,.CF]+CR+CAA

(8.10)

Eqn. (8.10) gives the total resistance coefficient of a ship in ideal condi tions. To allow for the actual conditions during ship trials or in service one may multiply the total resistance coefficient by a "load factor" (1 + x), the "overload fraction" x corresponding to a "trial allowance" or a "service al lowance" I ranging from 10 to 40 percent depending upon the type of ship and its service route. .

~.

i

I! '.~ j :1t.

185


I

I , I

For resistance experiments, it is first necessary to make an accurate ge ometrically similar model to an appropriate scale, which should be large enough for making the ship model and model propellers of sufficient accu racy and yet be within the limits imposed by the size of the towing tank and the speed of its towing system. Large models are also necessary to ensure turbulent flow around the model, and even then it is necessary to fit artificial turbulence stimulators at the bow of the model. The model is then accu rately ballasted to the required waterline and towed at a number of steady speeds in still water, the speeds covering slightly more than the normal op erating speed range of the ship. The tow force or resistance is measured by a resistance dynamometer at each speed, and the results analysed in a manner similar to that given in Example 1. In addition to speed and resis tance, it is also necessary to measure the teI!lperature of the towing tank water, s~ce the density and viscosity of water depend on its temperature. Resistance experiments may be carried out on a model without appendages such as rudders and shaft brackets, in which case one obtains' the "naked hull" resistance. The model may also be t~ted when fitted with appendages made to the same scale. If only a naked hull resistance test is carried out, it is necessary to add an empirical appen:dage resistance to the naked hull resistance.

8.3

Open Water Experiments

The object of an open water experiment :is' to determine the open water characteristics of a pr~peller. A geometrically: similar' model is made of the propeller, the size of the model propeller being~ebied!.by the size of the Bhi~el if it is intended to use the model propeller for sel~:p!.QP.u1Sion te~s also. If only open water tests are to be carried out, when generatingJ?.!~~ller method~~ata for example, the model P!~~leriSrii:8:& s~mewhat larger, its size depending upon the capacity of the propeller dynamometer available. In any case, model propellers should1be large :~nough!tQ:;ge made' accurately and to have a 'sufficiently high ReynoldsnUmberso ,thllot theflow is tll.¢ulent. A further measure to promote' turbulent' flow is"not to give ti;;model propeller a highly polished surface. It is re~o~ea'tliat' ;the model propeller ReynoldsIlu~b~~--based on-the resultanfofthe axial and tangential velocities and the section chord at 0.7 R be at -lea,;i-- 3.2 x 105 ; __

.

'

..-

7:.

_

'

•. _ , :

-----

l~

.~~


186

otherwise turbulence should be artificially stimulated, e.g. by roughening the leading edges of the blades. For the open water experiment, the model propeller is attached to a pro peller'dynamometer fitted in an "open water boat" as shown in Fig. 8.1. The propeller dynamometer measures the thrust and torque of the propeller. The propeller shaft extends a sufficient length forward from the boat to ensure that the flow around the propelle~ is 'not disturbed by the boat. A fairing cap IS 'provlded' a1 the forward end of the propeller boss. The open water boat is ballasted so that the propeller shaft is horizontal and its depth below the water surface is at least equal to the model propeller diameter. BALLAST , WEIGHTS

,MOTOR

FAIRING CAP,

DYNAMOMETER

~

~

~ 2~-=-::~-~ . '\ D b ~ ==/\

I

I

I

~~D

SHAFT TUBE

OPEN WAfER BOAT

~J

PROPELLER

o

Fiiure 8.1 : Open Water Experiment.

The open water experiment is conducted by towing the open water boat at a steady speed while running the propeller at a constant revolution rate. The speed of the boat (Le. speed of advance' ~), and the revolution rate n, thrust T and torque Q of the propeller are measured in each run. It is usuai to run the model propeller for all the speeds of advance at a constant revolution rate, this pr,eferably being the highest that can be attained at zero speed df advance at which the torque is maximum and which must be within the capacity of the driving motor and dynamometer. The speed of advance is varied in steps from zero to. the value at which the propeller thrust just becpmes negative. (Four quadrant measurements involving negative speeds and revolution rates are not considered here.) The measured thrust and torque are corrected for .the "idle" thrust and torque, i.e. the thrust and torque measured by the dynamometer when the experiment is carried out with a dummy boss of equal weight replacing the propeller. The open water characteristics of the model propeller can be easily cal culated from 'the measured values of ~ and n, and the corrected values of T and Q. The open water characteristics of the ship propeller 'will be


187

slightly different because of the difference between the Reynolds numbers of the model propeller and the ship propeller. The surface, of the ship propeller may also be rough. A procedure for correcting the open water characteristics for Reynolds number and roughness effects has been given in the ITTC 1978 performance prediction method. This procedure, which is based on the assumption that the blade section at O.75R can be assumed to represent the propeller as a whole, is summarized by the following equations:


188

-

nM

-

DM

= model propeller diameter

CM,CS

= expanded blade widths (chords) of the section at O.75R of the model propeller and the ship propeller respectively

lJM

= kinematic viscosi.ty of water for the model experiment

RncM

= Reynolds number of the model propeller

CDM,CDS

= drag

t C C

D ,,:i;;:'J:~

Z

kp

revolution rate of the model propeller I

i

~,

{ ~. ~ t \ I

~

coefficients of the blade section at 0.75R for the model propeller and the ship propeller respectively

=

thickness chord ratio of the blade section at O.75R

=

chord diameter ratio at O.75R

= =

number of blades propeller surface roughness (standard value 30 x 10- 6 m)

KTM,KTS

= thrust coefficients of the model propeller and the ship propeller respectively

KQM,KQs

= torque coefficients

of the model propeller and the ship propeller respectively.

Example 2 .. In an open water experiment with a model propeller of 200 mm diameter, a thrust of 252 N and a torque of 9.250 N m are measured with the propeller running at 2400rpm at a speed of advance of 5.6m per sec. The idle thrust and torque (to be subtracted from the measured values) are -4.0N and 0.034Nm respectively. Determine the thrust, torque and advance coefficients of the model propeller and its open water efficiency. The ship propeller has four blades, a diameter of 5.0 m and a constant pitch ratio of 0.8. The blade section at 0.75R has a thickness of 0.0675 m and a chord of 1.375 m. If the roughness of the propeller blade surface is 30 microns, determine the thrust and torque coefficients of the ship propeller and its open water efficiency.

, ?

Propulsion Model Experimen'ts DM nM

= =

189

200 mm = 0.200 m

TM = 252- (-4.0)

2400 rpm = 40.0 s-l

QM

'D s

==

5.0 m

kp

=

30fLm

J

=

Kn..f,

=

K Q .'o.I

=

P D = 0.8

=

=

Cs

= 9.250 -

=

256 N

0.034 = 9.216 N m

ts = 0.0675·m

1.375 mat 0.75R

30 x 10- 6 m

VAM

5.6 40.0 x 0.200

=

nMDM

TM PM n'i! D'1 QM

PMntD~

=

0.7000

=

256

1000 x 40.0 2 X 0.. 200 4

=

1000

X

9.216

40.0 2 X 0;2005

= 0.1000 = 0.01800

I

TJO.lo.,.f

CM

=

K M J -T- K Q M211"

= CDM sDs

=

0.1000 0.01800

= 1.375

X

0.. 7000, _ 0 X

~;

0.2~0 5.

=

-

8 .61 9

0.0550 m

' 2 = V 2 + (0.7511" nM DM) 2 = 5.62 VRM .+ (0.7511" x 40.0 AM

X

0.2) 2

= 386.6658 m 2 s-2 VRA.f

=

19.6638 ms

RncM

=

VRMCM VM

C

=

0.0675 1.375

CVM

=

2[1+2

t

= =

1

19.6638 x 0.0550 1.139 X 10- 6

= 0.949~ x 106

0 04909 .

n)] [R~cM 0.044

= 2 [1 + 2 x 0.04909] [

5 - RtM

]

0.044 ' ! (0.9495 x 106 ) 6

-

(0.9495

a]

5' X 10 6 ) 3

190


= 2 x 1.09818 (4.4381 x 10- 3 CDS

=2

[1+2

= 2 [1 +. 2 x 0.04909]

6.CD

=

CDS

p

6.KQ

-0.1562

= 8.6110 X

10- 3 )

10- 3

1 375 ] -2.5 1.89 + 1.62 log . 6 . 30 x 10

X

8.6110

C

= -0.3b..CD D

=

[

2 x 1.09818 [1.89 + 7.551:1.]-2.5

= CDM -

X

(~)] [1.89+1.6210g~:r2.5

.

=

0.5176

D Z

X

10- 3

= -

8.0195

8.0195

X

X

10- 3

10- 3 = 0.5915

= -0.3 x 0.5915 x 10- 3

X

10- 3

.

1.375 x 0.8 x - - x 4 5.0

10- 3

= 0.256.CD D·c Z

1.375 5.0

= 0.25 x 0.5915 x 10- 3 x - - x 4

= 0.1627 X 10- 3 KTS KQS 7]OS

8.4

= Kn[ = KQM -

=

0.1000 - ( -0.1562

X

10- 3 )

=

0.1002

6.KQ = 0.01800 - (0.1627

X

10- 3 )

=

0.01784

b..KT

KTS J =-- = KQs27l'

0.1002 x 0.7000 271' 0.01784

= 0.6256

Self-propulsion Experiments

Self-propulsion experiments are used to determine the performance of the ~ ~taken together. An analysis of the results of a self propulsion experiment 'allows one to predict the delivered power and the revoluJion rate of the ship propeller at a given speea of the ship, and to d~term~e-fraction,thrust deductiori'-fr~ction and rel~ive rota tive efficiency. The data of the resistanc~.and open water experiments are, however, required for analysing the data of the self-propulsion experiment.

.

....

if!, .


191

For a self-propulsion experiment, the model propeller is fitted in its cor rect position at the stern of the ship model and connected to a propeller dynamometer for measuring the thrust and torque of the propeller at vari ous revolution rates. The ship model should be fitted with all appendages as far as possible, particularly those lying in the propeller slipstream, e.g. a rud.de~~--The ship model is attached to a resi;t~nceaynamomete~ which i~- this test measures the f~~~~ ;~q~k~}Q:make-the ship model-,D1()ye _a:~_a constant speed with the pr~peller!unning. The shipIDQdeLis-ac.curately ballasted so that it float 13 at. the.correcLwaterline. The model is then towed at a st~~~eed with the propeller running at a constant revolution rate, and the thrust and torque of the propeller and the force appliegj!Q. the ship model thr;~gh-th~-;'~si~t~~e~ynamomet~r are~e~ld.r.~d. Fig. 8.2 ilh;:~t-;~ the set up. '.. experimental . . , ..... .... -_.- ...-.. ~

,~

1. RUDDER 2. PROPELLER 3. ,PROPELLER DYNAMOMETER 4. PROPULSION MOTOR 5. RESISTANCE DYNAMOMETER 6. BALLAST WEIGH TS

Figure 8.2: Self-Propulsion Experiment.

The analysis of the data recorded in a self-propulsion experiment is based on the following considerations. When a ship is moving at a steady speed Vs with its propeller producing a thrust Ts, then: RTS

= (I -

t) Ts

'-----~----

(8.12)

where RTS is the resistance of the ship at the speed Vs and t the thrust deduction fraction. If the model scale is A, and DM and Ds are the model and ship propeller diameters:

192

Basic Ship Propulsion (8.13)

Ds = DM>'

~he model must be tested at a speed VM such that the Froude numbers of the model and the ship' are equal, which implies that:

(8.14) If during the self-propulsion experiment at ,the speed VM the model pro peller;~runs at a revolution rate nM and has a thrust TM and torque QM while the force applied to the model:through·the resistance dynamometer is . I

F,

th~n:

F

= RTM -

! (1-

t) TM

i

(8.15)

RTM being the resistance of the model at the speed VM. This may bie written

as:

(8.16) whereKTBM is the thrust coefficient of the model propeller in the behind

condition, and PM the density of water in which the experiment is carried out.:Wth.~-m6~etpr~p;eIier'·and·theshippropeller are to fulfil the conditions of dYii~~'sii;{uaiitjr;' then:' ' . ;.

.~

,,'::;), J: .:, ' i .. ;"

~.' ~

!('

i . . ~:-::~:;}!:!

~;l.

,

. ". ~Js :l:::

'JM

KTBS = KTBM

(S.17)

tus - WM

whe~~' i;;'a:nd J M are the advance coefficients of the ship propeller and the ,tilb9:e1 :pr6'pel1~'r,· .. tL!S and tuM the corresponding wake fractions, and

c~efficie'nt' of the ship propeller in the behind condition. Using Eqns'iS.13) and (8.14). with the first equation of the Eqns. (8.17), one ; ' obtains:

KTBi th:e'thrUst

.V


193 (8.18)

, and then one may V{fite Eqn. (8.16) with the help of Eqns. (8.12) and (8.13) as follows: . F

2 4 PM RTM - (l-t)KTBSPsn s D s PS

DM )4 2(-D ) (-nM nS S

(8.19)

In terms pf the total resistance coefficients of the model a~d the ship, this becomes: 1 8 V2 F = C' TM 2PM M M

1

2

-

PM 1 ( ) C' 1 8 u2 \3 1 + x . TS'iPS S Ys PS 1\ I

-

(

S)2

V ) t' 2 Ss 1 +x GTS 'ij,PM8MVM 8 ( VM M

-

GTM2PM8MVM -

-

~PM 8M viI [GTM - ( 1 + x) GTS]

1 ,\3

(8.20)

Eqns. (8.19) and (8.20) show that for dynamic similarity as indicated by Eqn. (8.17), the ship model is not fully self-propelled, and the thrust of the model propeller must be augmented by the force F applied through the resistance dynamometer. The condition represented by Eqn. (8.19) or (8.20) is termed the "ship self-propulsion point on the model" in contrast to the "model self-propulsion point" at which F = 0 and the model is fully self-propelled by its propeller. It may be noted that whereas at the ship self propulsion point KTBM = KTBS, at the model self-propulsion point KTBM is greater than KTBs(at least for x = 0) so that the model propeller is overloaded in comparison to the ship propeller and is therefore operating less efficiently. It is possible for F to be negative if the overload fraction x is sufficiently large.

1----

194


If the self-propulsion experiment is to be carried out for only one value of the load factor (1 + x), then for each speed VM the value of the force F is calculated in advance using Eqns. (8.19) or (8.20), and with the model mov ing steadily with the speed liM the propeller revolution rate nM is adjusted until the calculated value of F is achieved. The thrust TM and the torque Q M are then measured along with VM, F and nM. This procedure may be repeated for several speeds to cover a range corresponding to the operating speed range of the ship. This method of conducting a self-propulsion exper iment is called the "constant loading" method or the "Continental" method because of its use in European ship model tanks. If self-propulsion ~est data are required for a different loading, the experiment is repeated for a new value of F at each speed. In an alternati\'e method of carrying out self-propulsion experinlents, called the "constant speed" method or sometimes the "British" method, tests are carried out for several values of nM at each value of VM, and F, T}.{ and Q AI are measured. This. allows the data to be analysed for different load factors (1 + x), which may be selected after the experiment is over. In actual practice, the constant loading method for different loadings' and the constant speed method for different speeds are equivalent since both lead to values of F. TM and QM as functions of both VM and n M . The analysis of the data of a self-propulsion experiment requires the re sistance of the model and the resistance of the ship derived from it, as well as the open water characteristics of the model propeller obtained through the open water experiment. The resistance of the model should be for the same model condition as in the self-propulsion experiment. This may involve correcting the model resistance data for any temperature difference between the resistance experiment and the self-propulsion experiment. The analysis then proceeds as described in the following. For each model speed VM and the corresponding ship speed Vs, the re quired force F is determined from the model resistance RTM and the ship resistance Rys using Eqn. (8.19). The model propeller revolution rate nM for this value of F~ and the corresponding values of TM and QM are then obtained, directly in the Continental method, and by interpolation if nec essary in the .British method. These are the values of nM, TAl and QM at the ship-propulsion point on the model for the speed VM. TM and QM are


J r

195

converted to the thrust and torque coefficients KTBM and KQBM for the behind condition:

I'

r

KQBM QM - 'PMn1-D~

KTBM

(8.21)

Referring to the open water characteristics of the model propeller (KTM , KQM and 170 as functions of J), one d€::termines:

(a) the values of J, KQM and 170 for KTM = KTBM, Le. for thrust identity (b) the values of J, KTM and 170 for KQM

= ~QBM' Le. for torque identity.

Then: w

-

1

JnMVM DM i

t

-

1- (PM RTS') / TM PS >.3 1-t

17H

- 1-.w

17R

-

17R =

[(QM KQBM KTBM KTM

(8.22) (thrust identity) (torque identity)

17D = 170 17R 17H

There are usually small differences between the values of w, 17H and 17R obtained using thrust identity anti the values obtained using torque identity, and the two sets of values are identified by the suffixes T and Q, e.g. WT and wQ are the wake fractions determined by thrust identity and torque identity


196

respectively. The values of t and TJD with thrust identity are the same as with torque identity. One may also determine the revolution rate and delivered power of the ship propeller from the following equations: Vs

VM >.0.5

nS

nM >.-0.5

PE

-

RTSVS

PD

-

27f PS n~ D~ KQBM

TJD =

(8.23)

PE PD

The values of T]D obtained by Eqns. (8.22) and (8.23) are identical. The analysis of self-propulsion experiment data may be carried mit using non-dimensional coefficients throughout. If the wake fraction, thrust deduc tion fraction and relative rotative efficiency are not required, the open water data of the model propeller are not required, and one may directly caJcu late the delivered power and revolution rate of the ship propeller from the self-propulsion experiment data using the Eqns. (8.23). The procedure for analysing self-propulsion experiments described in the the differences that exist between the model and the ship. These 9ifferences may cause the values of the delivered power and propeller revolution rate of the ship determined from the self-propulsion experiment to differ from the values obtained during the speed trials of the ship. A simple way to allow for these differences is to introduce correlation factors based on experience with previously built ships into the values predicted from self-propulsion experiments: forego~ng ignores

(8.24)

197


where k 1 and k2 are correlation factors for the propeller revolution rate and delivered power respectively. Example 3 The effective power of a single screw ship of length 100 m at a speed of 15.knots is 2150kW. The resistance of a 4.0m long model with appendages is 25.4N at the corresponding speed. Determine the tow force to be applied to the model in a self-propulsion test at this speed to obtain the ship self-propulsion point. The model propeller of diameter 0.2 m is found to run at '534 rpm for this tow force to be obtained, and the propeller thrust and torque are then 21.75 Nand 0.682 N m. The open water data of the model propeller are:

J

0.600

0.700 ;

0.800

KT

0.199

0.144

0.088

10KQ:

0.311

0;249

0.186

Analyse these data and determine the rpm of the ship propeller and the delivered power given that the correlation factors for rpm and delivered power are respectively 1.02 and 1.00. .

,\ == Ds

:=

Ls

:=

100 m

LM

:=

4m

D ....f

:=

0.2 m

Ls

:=

100

L ....,

Du,\

:=

:=

15 k

25.4 N

nM

:=

534 rpm == 8.9

21.75 N

QM == 0.682 N m

:=

2150 kW

RTM

=

TM

:=

:=

S-l

1 V.M == Vs,\ -0.5 == 7.:16 = 1.5432 ms-

25

4

== 0.2 x 25 == 5.0 m

RTS

PE

= V s

:=

25.4 _ 1.000 1.025 :=

7.716 ms- 1

Vs

PE

25.4 - 17.398.1

:=

X

2150 7.716

278.6418 kN

278.5418 x 1000 N 253

8.0019 N

PMRTS

t = 1-

'J.

~

PS

");3

TM

==

17.3981 21.75

1----

=

1 - 0.7999

:=

0.2001


198

p.\[n 2

M

D4M

= 1000 x 8.9 2 X 0.2 4 = 0.1716

D5M

= 1000 X 8.9 2 X 0.2 5 = 0.02691

0.682

QM

=

KQBM

21.75

TM

=

KTBM

2

PMnM

Torque Identity

Thrust Identity KT

=

= 0.1716

KTB.\[

J = 0.6500 7]0 -

WT

KTM KQM

KQ.\[

J

= 0.02801

=

0.6338

=

1- In}.[ D.\[

=

=

7]R

=

1-t 1-w

=

KQ.\[

0.7999 = 1.0670 0.7497

=

KQB.\[

0.02801 0.02691

=

7]H 7]0 7]R

=

1.0670 x 0.6338 x 1.0409

7]D

= 0.7039

0.6390

=

1

=

1 - 0.7703 = 0.2297

7]H

=

--

7]R

=

=

271"ps

'15

=

0.7999 0.7703

KTBM KTM

=

1.0384

=

0.1716 0.1618

= 1.0384 x 0.6390 x 1.0606

= 1.788- 1 = 106.8 rpm

n1 D~ KQBM =

271" x 1.025

3054.41 kW 7]D

0.6678 x 8.9 x 0.2 1.5432

= 0.7037 = 8.9 x

PD

0.1618

= 1.0606

1.0409 7]D

=

=

= 1 - 0.7497 = 0.2503 7]H

KT

0.02691

0.1618 0.6678 x- 0.02691 271"

wQ

1.5432

=

=

7]0

1- 0.6500 x 8.9 x 0.2

VM

= KQBM

J= 0.6678

0.1716 0.6500 0.02801 x ~

=

271"

KQ

2150 3054.41

=

0.7039

X

1.78 3 x 5.0 5 x 0.02691


199

Ship prediction:

ns

=

PiJ

= PD k2 = 3054.41 x 1.00

ns k1

=

106.8 x 1.02 = 108.9 rpm = 3054.41 kW

This neglects the differences between the open water characteristics of the model and ship propellers.

In the ITTC ship performance prediction method, a specific correction for the difference between the wake fractions of the ship and the model is introduced:·· i WTS

= (t

+ 0.04) + (WTM .

CVS t - 0.04)-C VM

(8.25)

where WTS and WTM are the wake fractions 9f the ship and the model, based on thrust identity, and Cvs and CVM are the viscous resistance coefficients given by:

CVS = (l+k)CFS+6.CF Cv M = (1 + k) CF M

(8.26)

The quantity 0.04 associated with the thrust deduCtion fraction t accounts for the thrust deduction contributed by a rudder in the propeller slipstream, and should be omitted if the rudder is not in the propeller slipstream, e.g. when there is· a single rudder on the centre line of a twin screw ship. The thrust deduction fractions of the ship and the model are assumed to be equal. Should Eqn. (8.25) yield a value of WTS greater than WTM, WTS should be taken equal to WT M . . The propeller loading at the ship self-propulsion point is then determined as follows: (1-

1-------

t ) Ts = RTS

200


that is, which reduces to: (8.27) where KTS is the thrust coefficient of the ship propeller in open water, and:

J TS

WTS) Vs = (1-nsDs

The value of KTS/ Jfs is first obtained from Eqn. (8.27). The values of JTS, KTS and KQs corresponding to this value of KTS/ Jfs are then determined from the open water characteristics of the ship propeller (Le. after correcting the characteristics of the model propeller for the difference in drag coefficients of the model propeller and the ship propeller.) The revolution rate of the ship propeller and the delivered power are then easily obtained:

ns

=

(1- WTS) Vs JTsDs

=

D 5 KQs 271" psns s -

(8.28) PD

3

TJR

\

Th~se

are values for the ship in conditions corresponding to the ideal con ditio~ of the model experi!llents. The propeller revolution rate n~ and delivered power PfJ for a ship during trials are obtained by applying corre lation factors or allowances, for which two alternative methods have been proposed: Method 1: Ph

=

PDCp

(8.29)

This is similar to the method described earlier. Method 2: ' In this, correlation allowances are applied to the resistance coefficient and the wake fraction of the ship for calculating the propeller loading: \.

Propulsion Model Experiments KTS

4s

=

201

Ss GTS + b.GFC 2 D~ (1 - t)( 1 - WTS + b.wc )2

(8.30)

This value is used to determine JTS, KTS and KQs from the open water characteristics of the ship propeller, and then: =

+

( 1 - WTS b.wC) Vs JTsDs

P!J = 271" PS n~ D~ KQS

(8.31)

'T]R

The correlation factors eN and Gp and the correlation allowances b..GFC and b.wc are obtained by analysing the data froin previous. model experiments and ship trials. The analysis of self-propulsion experiments described in the foregoing is for single screw ships. For twin screw ships, the analysis is identical·except that the resistance is shared equally by the two propellers and this must be taken into account. Self-propulsion experiments for ships with more than two propellers are complicated because there is an additional variable involved, viz. the distribution of loading among the different propellers.

8.5

Wake Measurements

Although the wake fraction is determined indirectly through a self-propulsion

test, it is desirable to observe the flow past a ship model at the position of the

propeller to determine the distribution of wake velocity over the propeller

disc directly. Wake measurements are also carried out to determine the radial

and circumferential distribution of wake velocity. Such measurements are

. useful for propeller design and for calculating unsteady propeller forces, and

could indicate unacceptable flow conditions at the stern, requiring changes

in the hull form of the ship. Direct methods of wake measurement make use of devices such as wake wheels and pitot tubes, Fig. 8.3, with a ship model in a towing tank or circulating water channeL These devices can only be used when the propeller is not fitted to the ship model, and hence they yield only the nominal wake


202

PilOT TUBE

!l

I 1

PilOT-STATIC TUBE

--$--@fg~-I

WAI
.

FIVE-HOlE SPHERICAL HEAD PI 10 r lUBE

Figure 8.3: Instruments for Wake Measurement.

(Sec.5.2). Other disadvantages of these devices are that their presenceaffects the flow around the ship model and that their ~esponse is slow. A..t!,t The wake wheel consists of a number of small vanes attached to radial spokes from a hub mounted on a shaft supported by very low friction bear ings. ';['he flow causes the wake wheel to revolve at a speed proportional to the flow velocity, and hence the average circumferential flow velocity at a given radius can be obtained. Repeated measurements with the vanes set at differ ent radii provide the radial variation of wake. The pitot tube measures the total (hydrostatic plus hydrodynamic) pressure at a point, and by subtract ing the hydrostatic pressure the hydrodynamic pressure is obtained which is proportional to the square of the flow velocity. Pitot-static tubes allow the total and the hydrostatic pressures to be measured together. By having a "wake rake" consisting of a series of pitot tubes mounted on a streamlined holder and connected to a manometer bank, wake measurements at several points can be obtained simultaneously. Pitot tubes however can be used only for the measurement of the axial velocity component of the flow. For obtaining all the three components of flow velocity at a point, it is necessary to use a five-hole spherical head pitot tube. Wake wheels, pitot tubes. and five-hole pitot tubes are calibrated by running them in undisturbed water

I I 1i

1


203

1.0

1.0

0.9

0.9

0.8

0.8 0.7

0.7

0.6

0.6 , 0.5

0.5

(

W (.

0.4

0.4

0.3

0.3

0.2

0.2 0.1 r) 0

0.1

6)

0 '-'--'--'--'--'-'-'--'--L--7-.L-I.-L-.L-I.-J-J....J

o

e

30 deg

60

90'

120

150

WI(

lao

I-L.....L...J'-'--'--'--'-.../..'-J....J

0

0.2 0.4 0.6 0.8 1.0

(

R

STN 1

I

1

•....

I

"-

I STN

0

•

"2

, "\ " , \ " \ "\ " "- ":'..j \

" <, "\ \ ", , ,

,

\ 45

,

0

\

\

I

~/

I

o·

I 0.61!.. 0.9R

l.OR 0,8R 0.6R 0.4R 0.2R

4'

/

/

/ (

10.7/ / / I 0.6 I .0.5 { I I 0.4

I

/

/ /

I== J!;=:

.' !

I

.

/ /

/

/

I

/

/ /

/

I I

/

/

/

I

4

/

/

/

J.

'2

/

/

.

,

1

0

r / /

0.3 / I .0.2

T

I

.·0.1

I I

.

135

I I

. I//t,,/

,.... -

Figure 8.4: Presentation ofWahe Data.

at known velocities. The disadvantages of these wake measurement devices are eliminated in laser doppler velocimeters, which allow velocity measure ments to be made without disturbing the flow, are very accurate and have.a quick response. Advanced laser doppler'velocimeters allow all three velocity components to be measured simultaneously. I '

The results of wake measurements may be presented in various forms, Fig. 8.4. Wake wheel measurements may, be used to determine the wake fraction at various radii, and the results presented in a form showing the


204

wake fraction as a function of radius. Pitot tube measurements may be pre sented in the form of diagrams showing the wake fraction at various points of the propeller disc at equal angular and radial spacing, or contours of con stant wake fraction. If all the components of wake velocity are determined, separate diagrams may be prepared for each component. Wake measurement data may be analysed to obtain the average wake fraction or the mean circumferential wake fraction at each radius. If v(r, e) is the axial velocity at any point (r, e) of the propeller disc, the average nominal wake fraction w is given by: (1 - w) V

7.

rn l Jor v (r, e) rde dr R

(R2 -

27r

=

rb

= 271"

l

R

r

(8~32)

v'(r) dr

rb

where V is the speed of the model, R the radius of the propeller" rb the boss radius, and v'(r) the axial velocity at the radius r averaged over the circumference (Le. the value that would be found by a wake wheel at the radius r). Alternatively, one may write:

_l

(l-w)VlI" (R 2 -rt)

R

rb

-l

r V [1_w(r,e)]rdedr Jo 21r

I

!

(8.33)

R2

71"V[1-w'(r)]rdr

rb

I \

where w(r, e) is the local wake fraction .at- the point (r, e) and w'(r) the average circumferential wake fraction at the radius r. The average wake fraction obtained in this way is the "volumetric mean" , Le. both sides of the Eqns. (8.32) and (8.33) represent the volume of water flowing through the propeller disc per unit time.

1

For studying unsteady propeller forces and propeller excited vibration, the local wake fraction w(r, e) may be expressed in the form of a Fourier Series:

!

n

W (

r, e) =

I:[am (r) cos me + bm (r) sin me ] m=O

(8.34)

I: !

I I


205

the coefficients am(r) and bm(r) being found by a least squares fit and the series being truncated after, say, twenty terms. Example 4 The local wake fractions obtained by measurements at various angular and radial positions at the location of the propeller in a single screw ship model are as follows: 8deg:

0

30

0.5832 0.5622 0.5328 0.4950 0.4489 0.3942 0.3312 0.2598 0.1800

0.4690 0.4521 0.4285 0.3980 0.3609 0.3170 0.2663 0.2089 0.1448

60

90

120

150

180

x=r/R 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.3013 0.2843 0.3524 0.4082 0.2905 0.2741 0.3397 0.3935 0.2753 0.2597 0.3219 0.3730 0.2558 0.2413 0.2991 0.3465 0.2319 0.2188 0.2712 0.3142 0.2037 0.1922 0.2381 0.2759 0.1711 0.1615 0.2001 0.2318 0.1342 0.1267 0.1570 0.1819 0.0930 0.0878 0.1088 0.1260 ./ Determine the average circumferential wake fraction at each radius and the average wake fraction over the propeller disc. 0.3426 0.3303 0.3130 0.2908 0.2637 0.2316 0.1946 0.1526 0.1058

.

For a single screw ship, the flow normally has port and starboard symmetry. The average circumferential wake at any fractional radius x = r / R:

w'(x) =

21r

r ~h

2..

w(x,B) dB = 2

X

2..

r w(x,B) dB

~h

For x = 0.2, using integration by Simpson's Rule:

B deg

w(r, 8)

8M

o

0.5832 0.4690 0.3426 0.3013 0.2843 0.3524 0.4082

1 4 2 4 2 4 1

30 60 90 120 150 180

f(w ' )

0.5832 1.8760 0.6852 1.2052 0.5686 1.4096 0.4082 \ 6.7360


206 1

1

1r

w' (0.2) = - x - x - x 6.7360 = 0.3742 11" 3 6

where 1/3 is the common multiplier in Simpson's Rule and 11"/6 = 30 deg is the spacing. The average circumferential wakes at the other radii are similarly obtained as:

w' (0.4)

= 0.3607 w'(0.6) = 0.2880 w'(0.9) = 0.1667

w'(0.3)

w'(0.7) w'(1.0)

w'(0.5)

= 0.3419 = 0.2529 = 0.1155

w'(0.8)

= 0.3176 = 0.2125

T,he average wake fraction over the whole propeller disc is given by: ·.2

( 1 - w) =

-~2

1-

In the present ca..e:e

xb

Xb

1·

1.0

"'b

2

•

-2 [1 - w' (x)] x dx = 1- 1- x b

1l.0

= 0.2, so that:

fl.o

2 1 fl.O w= 1-0.22 }O.2 w'(x)xdx = 0.48 }O.2 w'(x)x dx

Using integration by Simpson's Rule:

x -0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

w

w'(x)

xw'(x)

0.3742 0.3607 0.3419 0.3176 0.2880 0.2529 0.2125 0.1667 0.1155

0.07484 0.10821 0.13676 0.15880 0.17280 0.17703 0.17000 0.15003 0.11550

=

1

1

O. 8

3

SM

-1 4 2 4 2 4 2 4 1

f(w) 0.07484 0.43284 0.27352 0.63520 0.34560 0.70812 0.34000 0.60012 0.11550 3.52574

-4- x - x 0.1 x 3.52574 = 0.2448

f

w'(x) x ,dx

Xb


8.6

207

Cavitation Experiments

Cavitation experiments with propellers are carried out~o study propeller cavitation and determine its 'effects on propeller performance. In addition to observing cavitation patterns on propeller blades and the effect of cavita tion on the open water characteristics, such experiments allow one to study cavitation erosion and noise due to cavitation. Although it is possible to study cavitation on full size propellers in ships with the aid of underwater television cameras, cavitation experiments are usually conducted with model propellers in cavitation tunnels. There is also a depressurised towing tank at :MARIN, the Netherlands, which is a unique f~ility for studying cavitation. A variable pressure recirculating water tunnel, usually called a "cavitation tunnel" , Fig. 8.5, has a shell made up of a .number of pieges bolted together, an impeller to circulate the water in the tunnel, an.air pump to control the pressure in the tunnel, and the equipment aSsociated with the model whose cavitation characteristics are to be studied. The tunnel is normally erected in a vertical plane with the impeller at one: end of the lower horizontal limb, "* and the working section in which the model is placed in the upper limb. The flow is guided around the corners atthe junction~ of the horizontal and vertical limbs by guide vanes, and the rotation of the flow caused by the im peller is reduced to a great extent by a "honeycomb" section. A contraction section just' ahead of the working section increases the flow velocity to the desired value. A diffuser after the working section decelerates the flow and increases the pressure at the impeller position to minimise the possibility of impeller cavitation. A pressure coaming at the highest point of the tunnel is connected to an air chamber from which air may' be evacuated by a vac uum pump to allow the pressure of air above the water in the tunnel to be reduced. The pressure can also be increased to a high value to simulate the operating conditions of a submarine at great depth. For cavitation experiments with propellers, a model propeller is placed .in the working section and attached to a propeller dynarnoIl'leterj' which can measure the thrust and torque of the propeller while it runs at·a. steady revolution rate. The working section has large airtight windows. to allow cavitation to be observed with the help of stroboscopic lighting. Instruments are provided to measure the speed of water in the working section and the pressure. Non-uniform wake conditions ma:( be simulated by providing flow

L

208


3

Q

4

6

5

7

9

10

8

PROPELLER MOTOR AND DYNAMOMETER PRESSURE C~MING HONEYCOMB ~ CON fRACliON ./ 5. WORKING SE~T10N ./'

1. 2. 3. 4.

~

6.

DIFFUSER

7.

VERTICAL LIMB

8.

IMPELLER MOTOR

9.

IMPELLER

"-.../"

DIFFUSER.

..../

10.

if ---.-/'

\

Figure 8.5 : Cavitation Tunnel.

regulators, wire mesh screens or partial ship models ahead of the propeller. The non-uniform velocity field is measured by a wake rake and the flow regulating devices adjusted until the desired velocity field is obtained. It has been seen that cavitation is affected by the amount of air dissolved in water. An instrument to measure air content is therefore provided. How ever, air liberated from water by cavitation does not easily go back into solution and the air bubbles circulate with the water in the tunnel, interfer ing with the. cavitation phenomena and their observation. This problem is usually overcome by removing the air dissolved in the tunnel water. Alter natively, the tunnel may be provided with a "resorber" in which the,tunnel I

J. 1j

"r


209

water is circulated at low velocity and high pressure to force the air back into solution. The reduced air content and the purity of the tunnel watcrin comparison with sea water cause substantial differences between the cavita tion phenomena observed in a cavitation tunnel and those occurring wi~h}ull scale propellers at sea. Better correlation between the cavitation of ~. model propeller in a cavitation tunnel ano of the ship propeller at sea is obtained by reducing the cavitation number of the model propeller in comparison to the ship propeller by as much as 15-20 percent when air is rem()ved from the tunnel water. Cavitation nuclei may be artificially introduced by'elec trolysis in the tunnel water to increase its similarity with sea waterin'which microscopic particles act as nuclei. In carrying out propeller cavitation experiments in a cavitation tunnel, the thrust and torque of the model propeller are measured for differe l1t speeds of advance, revolution rates and pressures so'that the'·complete t:f!.p.ge· of advance coefficient and cavitation number is covered. In ad'dition, the t e.m7 perature of the tunnel water (on which vapour pressure depends) and its air content are also measured. Cavitation patterns are observed and recorded using powerful stroboscopic lighting and high speed photography. Noise measurements using hydrophones may also be carried out. Cavitation ero sion may be studied by covering the propeller blade su~faces with a suitable coating that is easily eroded by the collapsing cavities. The results of cavitation experiments are usually presented in terms of the thrust and torque coefficients as functions of the advance coeffident for different cavitation numbers, as shown in Fig. 6.4. Sketches showing the type and extent of cavitation for different advance coefficients and cavitation numbers or photographs may also be given. Zones of different types of cavitation may be shown in a diagram similar to Fig. 6.3. Cavitation noise and erosion parameters may be determined for given advance coefficients and cavitation numbers. In addition to cavitation experiments with model propellers, cavitation tunnels are also used to study cavitation on other bodies such as hydrofoils using appropriate instruments and equipment. The variable pressure towing tank of MARIN is 240 m long and 18 m wide, and has water up to a depth of 8 m. The towing carriage is driven by a cable towing system. The air pressure in the tank can be brought down to about 4kN per m 2 . The performance of the model propeller can be seen through a

J..

_

210


periscope by observers in a chamber in which normal atmospheric pressure is maintained and which is sealed off when the pressure in the tank is reduced. Example 5 A ship has a propeller of 4.0 m diameter with its centre line 3.0 m below the surface of water. The propelltlr in its design condition has a speed of advance of 8.0 m per sec at 150 rpm and 2500 kW delivered power. Experiments are to be carried out with a 1/16 scale model propeller in a cavitation tunnel in which the maximum speed of water in the working section is 10.0 m per sec, the maximum propeller rpm is 3000, the maximum torque is 45 N m and 'the minimum static pressure at the centre line of the working section is 15kN per m 2 . Determine the speed of water, the model propeller rpm and the static pressure at the propeller position if the model propeller is to be run (a) at the correct Froude number, and (b) just within the limits 9£ the cavitation tunnel. The tunnel water is deaerated and the tunnel cavitation number must be reduced by 15 percent coinpared to the ship. What is the minimum cavitation number of the tUl}Ilel based on the speed of water in the working section? D s = 4.0 m

ns

h s = 3.0 m

=

VAS

== 8.0 ms- 1

A - 16 DM = DS/A = 4.0/16 = 0.25 m

VAM max = 10.0 ms- 1 PTunnel

== 2.5 s-l

150 rpm

max = 3000 rpm = 50 s-l

nM

PDS UM

QM

= 2500 kW = 0.85us

max = 45 Nm

min = 15 kN m- 2

\,

JS = =

Us

==

V:4S

nsDs

=

8.0

PDS

271 psn~ D~ PA

= 0.800

2.5 x 4.0 =

+ Psghs ~ps lf1s

2500 == 0.02426 271 x 1.025 X 2.5 3 x 4.0 5 PV

=

101.325 + 1.025 x 9.81 x 3.0 - 1.704 ~ x 1.025 X 8.0 2

== 3.9569 (a) At the correct Froude number

VAAl

=

F AS A- 0 . 5 = 8.0

X

16- 0 ,5

=

2.0 ms- 1

.'

.~.

211 ..


=

2.0 0.800 x 0.25

PTunnei -

PTunnel - PV

1.704

= z1 x 1.000 X 2.0 2

V2

1

= 10.0 S-l

ZPM AM

= 0.850"s

= 0.85 x 3.9569 = 3.3634 PTunnnel

=

1

.

1.704 + 2 x 1.000

X

2.02 x 3.3634

.

= 8.4307 kN m

2

(which is below the minimum limit of pressure)

(b) At the limits of the cavitation tunnel

nM

=

= 1.000

=

X

50.02

X

0.25 5 x 0.02426

(KQM

=. KQs)

0.05923 kN m = 59.23 N m, which is above the limit.

Hence, keeping Q M

=

10.0. 0.800 x 0.25

= 45 Nm, the limiting value,

QM PMDLKQM

=

1000

X

45.0 = 1899.4229 0.25 2 x 0.02426

nM

=

43.5824 s-l

VAM

=

JMnMDM = 0.800 x 43.5824 x 0.25

PTunnel

=

PV

+ ~PM V1 M O"M =

= 8.7165ms- 1

1.704 + ~ x LOOO 2

X

8.7165 2 x 3.3634

= 129.475 kNm- 2

The minimum cavitation number of the tunnel is: min 15.0 - 1.704 0" - PTunnel - PV = = 0.2659 T 1 vmax2 1 zPM AM 2 x 1.000 X 10.02


212

Problems 1.

A resistance experiment is carried out on a 5.0 m long model of a ship of length 120 m and wetted surface 3200 m 2 • The following results are obtained: Model speed. m per sec

1.050

1.260

1.470

1.680

1.890

Model resistance, N

12.13

17.78

. 25.45

36.29

52.02

Determine the effective power of the ship at the corresponding speeds using the ITTC friction line with a form factor of 1.06 and a correlation allowance of 0.0004. 2. A ship of length 150 m has a transverse projected area above water of 403 m 2 , the wetted surface of the hull being 6200 m 2 • The bilge keels have a wetted surface of 260 m 2 • The roughness of the hull surface is 150 microns. ;The resistance of a model of length 6.0 m as measured during a resistance test is as follows: Model speed, m per sec

1.235

1.440

1.646

1.852

2.058

Model resistance, N

26.99

38.45

54.54

77.71

111.56

Determine the effective power of the ship at the corresponding speeds using the ITTC performance prediction method with a form factor of 1.05. 3. An open water experiment covering the complete range of advance coefficient is to be carried out on a model of a propeller of diameter, 5.0 m and pitch ratio 1.0. The effective pitch ratio of the propeller is expected to be about 1.08, and the torque coefficient at zero speed of advance to be 0.0625. The maximum " speed at which the model propeller can be made to advance is 6.0 m per sec, and the maximum rpm and torque of the propeller dynamometer are 3000 and 3.0 N m respectively. Determine the maximum model propeller diameter if (a) the model propeller is to run at a constant 3000 rpm for the complete range of advance coefficient, and (b) the model propeller is to run at 2400 rpm. If the minimum propeller diameter is to be 150 mm, what is the maximum rpm? Calculate for each case at zero advance and zero slip the Reynolds number of the model propeller using the resultant velocity at 0.75R, given that the chord diameter ratio is 0.28 for this radius. 4. An open water experiment is carried out with a four-bladed model propeller of 200 mm diameter and 0.8 pitch ratio. The blade section at 0.75R has a chord of 50 mm and a thickness of 2.5 mm. The model propeller is run at 2100 rpm over a range of speeds and the following values of thrust and torque (corrected for idle thrust and torque) are obtained:

~

i

213

Propulsion Model Experiments Speed, m per sec

0.000

0.700

1.400

2.100

2.800

666

612

551

484

412

Torque, Nm

15.29

14.54

13.57

12.38

10.98

Speed, m per sec

3.500

4.200

4.900

5.600

6.300

Thrust, N

333

247

156

59

-45

Torque, Nm

9.35

7.50

5.43

3.14

0.62

Thrust, N

Determine the open water characteristics of the full size propeller of diameter 4.0 m, assuming that its surface roughness is 30 microns. 5. In a self-propulsion test, the model speed is 2.0 m per sec and at the ship self-propulsion point the 0.2 m diameter model propeller runs at 900 rpm and the thrust, torque and tow force are respectively 55.00N, 1.33 Nm and 2.55N. The resistance of the model at 2.0 m per sec is 49.03 N. The model propeller has open characteristics as follows: 0.300

0.400

0.500

0.600

0.700

0.210

0.182

0.lq2

0.120

0.086

0.253

0.225

0.195

0.164

0.132

Calculate the propulsive efficiency and its components on the basis of (a) thrust identity, and (b) torque identity. If the model scale is 1:25, de termine the effective power, the delivered power and the propeller rpm of the ship at the corresponding speed. Neglect the difference between the open water characteristics of the model and ship propellers. 6. In a self-propulsion test with a twin screw ship model, the model is self propelled at a speed of 2.3 m per sec when the two propellers together have· a total thrust of 34N and a total torque of 1.18Nm at 720 rpm. In the open water test with either propeller running at 720 rpm a thrust of 17 N is obtained at a speed of 2.0 m per sec, the torque being 0.61 N m. The resistance of the model at a speed of 2.3 m per sec is 29 N. Calculate the propulsive efficiency and its components. Are these values applicable to the ship? 7. A ship of length 100 m and a wetted surface of 2250 m 2 has a propeller of diameter 5.0 m. At its design speed of 20 knots, the ship has an effective power of 5000 kW. A self-propulsion test is carried out on a 1/25-scale model, which has a resistance of 35 N at the corresponding speed. The following readings (after correction) are obtained at this speed:


214 500

550

600

650

700

27.15

30.84

35.83

42.22

50.81

Torque, :\m

0.75

0.93

1.42

2.08

3.10

Tow Force, N

6.33

5.47

4.65

3.70

2.63

Propeller rpm Thrust, :\

The open water data of the model propeller are as follows:

J

0.700

0.800

0.900

1.000

.0.255

0.224

0.191

0.153

0.508

0.468

0.420

0.368

Calculate the delivered power of the ship and its propeller rpm at 20 knots and the propulsion factors using the ITTC ship prediction method with the following values: 1 -+- k

D..J{Q

= 1.05 = 0.0003

D..CF = 0.6 CN

10- 3

X

D..KT = -0.0002

= 1.005

Cp = 1.010

8. In a model of a single-screw ship, the following average velocities over the circumference at various radii at the propeller position are measured with the 1p.odel moving at a speed of 5.0 m per sec in undisturbed water: Radius, mm Velocity, m per sec Radius, mm Velocity, m per sec

20

30

40

50

60

3.503

3.557

3.632

3.730

3.848

70

80

90

100

3.988

4.150

4.333

4.538

Determine the nominal wake fraction, the model propeller having a diameter of 200 mm and a boss diameter ratio of 0.2. 9.

In a twin 'screw ship model, the axial velocities measured at various radii r and angular positions e at the propeller disc when the model has a speed of 4.0 m per sec are as follows: •

Propulsion Model Experiments 0°:

0

rmm

0°:

15.0 22.5 30.0 37.5 45.0 52.5 60.0 67.5 75.0

30

60

90

120

150

Axial velocity, v m per sec

15.0 22.5 30.0 37.5 45.0 52.5 60.0 67.5 75.0

rmm

215

3.600 3.552 3.488 3.440 3.368 3.308 3.228 3.136 3.040

3.680 3.636 3.596 3.552 3.508 3.448 3.400 3.348 3.280

3.792 3.788 3.764 3.760 3.752 3.728 3.716 3.6,96 3.680

3.880 3.880 3.876 3.876 3.872 3.868 3.852 3.840 3.828

3.960 3.956 3.952 3.952 3.948 3.948 3.940 3.928 3.920

3.960 3.956 3.948 3.936 3.924 3.916 3.892 3.872 3.840

180

' 210

240

270

300

330

Axial velocity, 1,1 m per sec 3.936 ,3.880 3.928 3.872 3.920 3.856 3.896 3.836 3.880 3.808 3.864 3.776 3.840 3.740 '3.800 3.700 3.756 3.656

3.844 3.836 3.816 3.792/ 3.760' 3.724 3.680 3.632 3.584

3.640 ,3.608 3.568 3.528 3.484 3.440 3.392 3.340 3.280

3.440 3.408 3.312 3.236 3.160 3.880 2.996 2.896 2.800

3.200 3.120 3.032 2.936 2.844 2.744 2.640 2.520 2.348

Determine the mean circumferential wake fraction at the various radii and the nominal wake fraction. The model propeller is of 150 mm diameter with a boss diameter of 30 mm. 10. A single-screw ship has a design speed of 22 knots with its propeller run ning at 150rpm and the delivered power being 12500kW. The propeller has a diameter of 6.0 m and its centre line is 5.2 m below the water line. Model experiments are to be carried out in a cavitation tunnel with a free surface working section using a ship model to create a variable wake. The model propeller diameter is 150 mm. Determine the speed of water in the working section ahead of the model, the model propeller rpm and torque, and the pres sure in the air above the water surface in the working section for simulating the conditions in the ship propeller. Assume that the cavitation number of the model propeller must be 10 percent less than that of the ship propeller.

L-------------

CHAPTER

9

Propeller Design 9.1

Propeller Design Approaches

Propellers for ships may be designed to fulfil two purposes: (i) to utilise the available propelling power efficiently to propel the ship alone, or (ii) to use the power to allow the ship to tow other vesst:ls or gear. Propellers designed for propelling the ship alone are called "free running propellers". PJ;"opellers fitted to tugs, towboats and trawlers are called "towing duty propellers". The design of propellers can be approached in two different ways. In the first approach, propeller design is based on methodical series data so that many design features such as the blade outline and blade section shapes are fixed by the methodical series adopted, and the design of a propeller reduces to determining the diameter, the pitch ratio and the blade area ratio. This appfoach has been widely used for several decades and is even now often adopted for designing propellers which are moderately loaded and in which cavitation is not a major problem. Even when a design method using the other approach is to be adopted, a design obtained from a methodical series often forms the starting point. The second approach to propeller design is based on· the use of propeller theory. A suitable distribution of circulation required to give the specified thrust is determined, and then the detailed design of the blade sections at a number of radii is carried out so that the required circulation is obtained while at the same time the minimum pressures are kept within safe limits to avoid harII).ful cavitation. A major advantage of this approach is that the

216

I ~

Propeller Design

217

blade sections can be designed to suit the average radial wake variation in the flow incident on the propeller. The calculations in the design approach based on propeller theory are somewhat involved but can now be routinely " carried out with the help of computers.

9.2

General Considerations in I>ropeller Design

Before discussing propeller design procedures in detail, it is useful to exam ine the considerations involved in selecting the 'major design features of a propeller. These consideratio'us often form ancimportant part of the pro peller design proce~s,· and in many cases necessitate a compromise between several conflicting requirements. The propeller revolution rate (rpm) is usually determined by the propul sion plant that is selected for a given ship. However, in deciding upon the propeller rpm it is necessary to consider several fadors related to propeller design. It is important to select the propeller rpm so that resonance with the natural frequencies of vibration of the hull,~nd the propeller shafting system are avoided. An unduly high propeller rpm may incr~ase susceptibility to cavitation. On the other hand, a low propeller rpm results in a large propeller diameter (other things being equal) and an increase in efficiency. This has resulted in recent years in the development of very low speed diesel engines with speeds as low as 50-60 rpm, which in association with large diameter propellers have led to substantial improvements in propeller efficiency. From the point of view of vibration, propeller rpm must be considered in association with the number of propeller blades. The propeller excited frequ~ncy of vibration is the product of the propeller rpm and the number of blades and it is this product which must lie well clear of the natural /~frequencies of hull vibration (vertical, horizontal and torsional) as well as the natural frequencies of the propulsion shafting system. One may also note that the larger the number of blades the smaller is the exciting force per blade. On the other hand, the smaller the number of blades the greater ;" is the optimum propeller diameter and higher the propeller efficiency, but there is also an increase in the weight of the propeller. _ The diameter of the propeller is influenced by the propeller rpm, a lower rpm resulting in a higher optimum diameter and a higher efficiency. The

218


maximum propeller diameter is limited by the need to maintain adequate clearances between the propeller and the hull and rudder. Typical values of the minimum clearances required by a classification society (Lloyd's Register / of Shipping) are given in Fig. 9.1. While a larger propeller diameter reduces the possibility of cavitation by reducing the propeUer loading, it could lead / to excessive wake \-ariations and. to a reduction in propulsive efficiency due to a decrease in the average value' of the wake fraction.

tI

r

The pitch ratio of a propeller governs the power that it will absorb in given; operating conditions, i.e. speed of advance and rpm: the higher the pitch ratio the higher the delivered power at a cons~ant advance coefficient. The effect of pitch ratio on propeller rpm is given approximately by the following empirical relationships due to van Manen (1957):

8n/n - 1 8(P/D)/(P/D)

at constant torque

8n/n 8(P/D)/{P/D)

at constant power

P

+D =

constant

1.5

(9.1)

at constant power and rpm

These relations may be used to make small corrections to the pitch and diameter of an existing propeller design in which the relationship between po,,'er and rpm is not fully satisfactory. The blade area ratio is selected basically from considerations of cavitation, a certain minimum blade area being necessary to ensure that even if cavi tation occurs it is within acceptable limits. Unduly large blade area ratiosJ cause a reduction in propeller efficiency due to an increase in blade section drag. Very low expanded blade area ratios (below about 0.3) may lead to difficulties in generating adequate astern thrust. A small increase in blade / area ratio may be made to keep propeller blade stresses within permissible limits without an increase in the blade thickness fraction. The boss diameter ratio of fixed pitch propellers usually lies between 0.15 and 0.20, values outside this range resulting in a drop in propeller efficiency..~ The minimum permissible boss diameter, however, depends upon the diam- / eter of the propeller shaft. .

J

! j

.J

j

·

I t~

,

'.;.:

.

Propeller Design

219·

I

I' I

I e

(b) TWIN SCREW Number of Blades

a

b

c

e

3

1.20 KO

1.80 K 0

0.120

o'Or O

1.20 KO

1.20KO

4

1.00 KO

1.50 K 0

0.120

0.030

1.00 KO

1.20 KO

5

0.85KO

1.275KO

0.120

0.030

0.85KO

0.85KO

6

0.75KO

1:125KO

0.120

/0.030

0.75KO

0.75KO

Minimum Value

0.100

0.15D

0.200(1)

0.15D

d

.,

O.16D~I)

.

.~.-.-

L

:=

(i) for 3 or 4

blades

(ii) for 5 or 6

blades

Length of the ship in m

Cs = Block coefficient at load drought Ps Designed power on one shaft In kW t

:=

Thickness or rudder in m measured at 0.7R above the propeller shart centre line

R

Propeller radius in m measured at 0.7R

o

Propeller diameter in m

Figure 9.1 : Propeller-Hull Clearances.

~--~---

220


Propeller blades are sometimes raked aft to increase the clearances be tween the hull and the propeller blade tips and leading edges. This allows a,/ propeller of a larger diameter to be used, and results in a higher efficiency provided that the propeller rpm can be suitably selected. Aft rake, how ever, causes an increase in the bending moment due to the centrifugal force ./ on each blade, thus requiring thicker blades wbich have a lower efficiency. Slow running propellers may be given a rake aft of up to 15 degrees, but in propp.llers running at high rpms aft rake is best avoided.../ Propellers in which the blades are skewed back (Le. towards the trailing edge) result in a lower magnitude of unsteady forces generated by the pro peller in a circumferentially varying wake. Skewed blades are also believed " to be less liable to get entangled with ropes or chains that accidentally comeJ in way of the propeller. However, propellers with heavily skewed blades j have low backing efficiencies, are difficult to manufacture and require special strength considerations. In the foregoing, the factors to be considered in selecting the overall pro peller design parameters, viz. number of blades, diameter, pitch ratio, blade area ratio, boss diameter ratio, rake and skew, have been discussed. Some of these parameters can be selected by a designer using the methodical se ries method of propeller design. When a design method based on propeller theory is used, some additional parameters must be considered. The radial distribution of loading (i.e. the variation of circulation with radius upon which the radial distribution of thrust depends) is normally made' optimum for the given average wake a.t each radius. In some cases, however, the loading may be decreased towards the blade tips to reduce cavitation, blade stresses and propeller induced hull vibration. Departures from the optimum load distribution are naturally accompanied by a loss in efficiency. The shape of the expanded blade outline is chosen in accordance with the radial load distribution, the higher the load the greater the blade width,/' required at that radius to ensure that cavitation is kept within limits. Narrow blade tips result in an increase in propeller efficiency but also a greater risk of harmful cavitation. The shape of the blade outline for a given expanded area also depends upon how the skew is distributed along the radius. The radial distribution of skew (Le. the shape of the line joining the midpoints of the blade widths at the different radii) affects the unsteady forces generated

Propeller Design

221

by the propeller blades in a variable wake field, and it is possible to choose the shape of the blade outline so as to minimise these unsteady forces. The efficiency of a propeller depends upon the shape of its blade sections. Aerofoil sections with their high lift-drag ratios result in high efficiencies. However, such sections also have a high suction pressure peak and are thus more prone to cavitation. Segmental blade sections have a somewhat lower lift-drag ratio but a more uniform pressure distribution making them less efficient but also less liable to cavitation. Propellers are therefore often designed to have aerofoil sections at the inner radii where cavitation is less likely. and segmental sections towards the blade tip where back cavitation usua.ny starts. For propellers in which cavitati~n is likely to be a problem, special sections such as the Karman-Trefftz section and the NACA-16 and NACA-66 sections with a = 0.8 and a = 1.0 rdean lines are used. Details of these NA;CA sections are given in Appendix 2. .

,

The blade· section thickness is governed mostly by the strength require ments of the propeller. A lower thickness-chord ratio reduces drag and in creases propeller efficiency but also reduce~ the range of angles of attack in . which no cavitation occurs. The thickness~ndchord of the blade sections at different radii must therefore be based on loading, strength, cavitation and efficiency considerations. The blade section camber depends upon the loading at each ra.dius and must be such that in association with the angle of attack it results in the required lift coefficient. The angle of attack depends upon the direction of the resultant velocity and the pitch angle of the blade section at a particular radius. The pitch angle should be chosen so that the ideal angle of attack is obtained. However, a propeller normally works in a variable wake, and it is necessary to choose the pitch angle so that back cavitation due to high angles of attack and face cavitation due to low (negative) angles of attack are both avoided in the given flow field.

\

From the foregoing discussion, it is evident that in common with most design processes, propeller design involves a number of mutually conflicting considerations, and compromises must often be made in the pursuit of an optimum design. It may be added that specific decisions regarding each and every design variable discussed in this section are neither necessary nor possible in normal propeller design calculations. These general design

\

L

222


considerations merely serve as a background to the methods used in propeller design.

9.3

Propeller Design using Methodical Series Data

Methodical series data may be used to design both free running and towing duty propellers. The design of towing duty propellers is considered in the next section. The first decision to be made in propeller design based on me thodical series data is the choice of the methodical series. This decision may be based on the considerations discussed in the previous section taking into account the particular features of the different methodical series available. The MARIN B-Series is widely used for propeller design because experien~e has shown that it has excellent performance characteristics, particularly for moderate loadings. For heavily loaded propellers used in high speed, twin screw ships, the Gawn series may be preferred because it has segmental blade sections which are less likely to cavitate. Propeller methodical series data are available in the form of design charts or as regression equations suitable for use with a computer. The most con venient type of propeller design chart is the Bp - 0 diagram, or its modern variant in which Bp is replaced by KQI J5 and 6 by J. Design problems for free running propellers are of two types: (i) Given the propeller diameter and the ship speed, design the optimum propeller to minimise the power required, and determine the corre sponding propeller rpm. (ii) Given the engine power and propeller rpm, design the optimum pro peller to maximise the ship speed. The first type of problem basically provides guidance in selecting the propul sion plant for a ship. The second type of problem yields a propeller design to suit a given pro~ulsion plant, which is selected considering several factors in addition to the requirements of the propeller. Incidentally, if one first starts with a given propeller diameter and ship speed and determines the opti mum propeller rpm [type (i) problem], and then takes this rpm as given and determines the optimum diameter [type (ii) problem], one obtains a higher

223

Propeller Design

diameter than the initial value. If this process is repeated, one gets higher and higher values of diameter and correspondingly lower values of propeller rpm, keeping the ship speed constant. The data required for propeller design include the following: (a) Effective power for a

r~nge

of speeds, [one speed for a type (i) problem].

(b) The propulsion factors - wake fraction, thrust deduction relative rotative efficiency.

f~action

and

(c) The depth of immersion of the propeller axis. (d) The shafting efficiency. (e) The ship speed and the propeller diameter for a type (i) design problem or the engine power and rpm and the gear ratio (engine rpm: propeller rpm) if gearing is provided between the engine and the propeller, for a type (ii) design problem. Individual propeller designers have their bwn procedures for designing pro pellers. Typical design procedures for propeller design using methodical se ries design charts (or data extracted from them) and regression equations are illustrated in the following examples. The methodical series data used in these examples are given in Tables 9.1 and 9.2. It is important to note that these data are only for illustrative purposes and are not based on the data of an actual methodical propeller series. The data in these tables should be used only in the following ranges of pitch ratio and advance coefficient: 0.7

~

P D

~,1.1

p ( D

0.3) <- J <- DP

Example 1 A ship is to ha.ve a. design speed of 16 knots at which its effective power is 3388 kW. The maximum propeller diameter that can be fitted is 5.5 m. The propulsion 0.150, 1JR 1.050. The factors based on thrust identity are: w = 0.200, t shafting efficiency may be taken as 0.970. The minimum blade area ratio required to keep cavitation within acceptable limits is estimated to be 0.55. Determine the

=

=

~ ~

Table 9.1

,;:..

Optimum,Efficiency Line for 4-Bladed Propellers·

(KQ)! . J5

~~

AE

-

= 0.40

Ao

.J

P D

TIn

.T

0.5000

0.8679

1.0999

0.7111

0.8625

0.5500

0.7802

0.9968

0.6926

0.6000

0.7161

0.9320

0.6500

0.6647

0.7000

~~

= 0.55

p

= 0.70 ~

TIn

.T

1.1000

0.7084

0.8614

1.1002

0.6989·

0.7838

1.0141

0.6886

0.7894

1.0350

0.6848

0.6763

0.7180

0.9445

0.6720

0.7201

0.9567

0.6678

0.8840

0.6607

0.6671

0.8979

0.6562

0.6690

0.9091

0.6519

0.6220

0.8482

0.6453

0.6249

0.8625

0.6407

0.6276

0.8753

0.6362

0.7500

0.5854

0.8184

0.6303

0.5889

0.8343

0.6254

0.5922

0.8483

0.6208

0.8000

0.5533

0.7930

0.6156

0.5S"{4

0.8104

0.6104

0.5613

0.8260

0.6057

0.8500

0.5249

0.7708

0.6014

0.5297

0.7905

0.5959

0.5339

0.8069

0.5909

OJ

0.9000

0.4998

0.7523

0.5876

0.5049

0.7727

0.5818

0.5096

0.7909

0.5766

n

0.9500

0.4770

0.7348

0.5743

0.4827

0.7578

0.5682

0.4876

0.7766

0.5628

1.0000

0.4565

0.7199

0.5616

0.4624

0.7438

0.5551

0.4677

0.7640

0.5495

D

D

• These data may be obtained from diagrams similar to Fig.4.l).

rIo

~

....

~ '6" ~

.g I::

.... 0

til

t:l

~-if:t~::..,"'''-''----

.

_.,

--"._-'-"''''-<

_.

. . . - - - - - - - ' - -.. - - . . . - - - -

-~'f.~f''Q··-ttMC'= 5

m -t"

·f"~.

225

Propeller Design Table 9.2 Regression Equations for Thrust and Torque Coefficients of 4-Bladed Propellers

KT - 0.446 - 0.3130 J

P+ 0.3447 D

P

0.0315 J D

+ 0.0495 (P)2 D

AE PAE Ao + 0.2823 J Ao + 0.8590 D Ao (DP)2 - 0.3844 AE - 0.9800 J -P-AE - 0.3533 (P)2 -AE + 0.5000 J (P)2 -AE

- 0.0100 J

D Ao

lOKQ

=,0.0391 + 0.1664 J -

D

'

D

Ao

P

Ao

, P' (P)2 + 0.6817. D

0.0881 D - 0.6128 J D

AE PAE Ao + 0.1533 J Ao + 0.1352 D Ao (DP)2 - 0.1092 AE PAE (P)2 AE . (P)2 -AE - 0.6667 J - - + 0.2183 - + 0.1333 J -

+ 0.0050 J

D Ao

D

D

Ao

Ao

\

optimum propeller rpm, the corresponding brake power and the pitch ratio of the propeller. (a)

Solution using Optimum Efficiency Line data, Table 9.1

v = 16knots D

=

5.5 m

= 8.2304ms- 1

PE = 3388kW

w = 0.200 t = 0.150 1JR

=

1.050 1Js

VA = (1- w) V = (1- 0.200) x 8.2304

J =

~

nD

=

6.5843 n X 5.5

or

,n

= 0.970 AE = Ao

= 6.5843ms- 1

-1 = 1.19715 J s

0.550

226

Basic Ship Propulsion 1- t

=

1 -: 0.150

1 _ 0.200

=

1JH

= 1_ w

PD

= 2'iT{Jn3 D 5 KQ/T/R = 211" x 1.025 x n 3 X 5.55 K Q/1.050kW =

1.0625

30869 n 3 K Q kW

PD T/O T/H T/R ;:::: PD T/O

x 1.0625 x 1.050 = 1.11562 PD T/Q

From Optimum T/o line for

AE/Ao =0.55

(~~Y

P

J

D

0.8625 0.7838 0.7180 0.6671 0.6249 0.5889

0.5000 0.5500 0.6000 0.6500 0.7000 0.7500

1.1000

0.708~

~.0141

0.6886 0.6720 0.6562 0.6407 0.6254

0.9445 0.8979 0.8625 0.8343

[KQ from (KQ fJ5) ~ and

/

,-,/

10KQ

T/o

0.2983 0.2707 0.2473 0.2358 0.2288 0.2241

n

PD

PD 1J0 T/R 1JH

s-l

kW

kW

1.3880 1.5274 1.6673 1.7946 1.9157 2.0329

2462 2977 3538 4207 4966 5812

1946 2287 2653 3080 35409 4055

J]

At PD 7J0 1JR TlH ;:::: PE ;:::: 3388 kW, by linear interpolation:

n = 1.87415- 1 PD

P D

(b)

;::::

= 4705kW

=

112.4rpm PE

=

PD

Tis

=

4705 0.970

=

4851 kW

0.8747

Solution using Regression Equations, Table 9.2

As Ao

KT

;:::: 0.55

=

Hence, substituting this ·value in the equations in Table 9.2:

-0.1,668 - 0.1577 J

P + 0.8172 D

~0.2650J (~r

P 0.1448 -- 0.5705 J D·-

(P)2 D

~.'

'J

Propeller Design

227

P+ 0.8018 (P)2 D

p

10KQ = -0.0210 + 0.2507) - 0.0137 D - 0.9795) D

. (P)2 + 0.0783) D

R

KT

J2

= =

=

1-t pD2(1~w)2V2

=

1

pD2(1-t)(1-w)2 V 3

1. 3388 = 0.3603 1.025 x 5.5 2 x ( 1 - 0.150)( 1 - 0.200)2 8.23043

The pitch ratio has to be determined that will give the highest efficiency at this value of .KTjJ2.0ne may proceed as follows: . .,

P D = 0.8 .

KT = 0.39429 - 0.44450 )

10KQ = 0.48119 - 0.48279 )

~; = 0.39429G)' - 0.4445l~ = 0.3603 1

) = 1.67341 K T = 0.12857 _ KT ) _ KQ 27f -

Tlo -

) = 0.59758 10KQ = 0.19268

0.12867 0.59758= 0.63512 0.019268 x . 27f

Proceeding similarly for other pitch ratios until the values of efficiency converge to the optimum, one obtains:

228


,~,

n

=

PD = PB P D

=

=

JD

6.5843 0.70015 x 5.5

PE

=

170 1]R 1]H

PD

-

178

4663

=0.970

=

1.7098s- 1

102.6rpm

3388 = 4663kW 0.65118 x 1.050 x 1.0625 = 4808kW

= 0.9906

The differences' between the answers obtained by the two methods hrise because of .the difference between the definitions of the optimum efficiency in the two cases: in the method using the data of Table 9.1, the optimum efficiency is calculated for constant values of KQ/ J 5 , whereas in using the regression equations, the optimum efficiency is calculated for a constant value of KT/ J2. The latter is theoretically more correct for this type of problem, but most methodical series design charts give only the optimum efficiency line for KQ/J5 (or Bp) constant. The error at least so far as power is concerned is small, and since the object in this problem is to obtain an estimate of the power for selecting the propulsion plant, this error is usually acceptable. Example 2

The effective powers (naked hull) at different speeds of a single-screw ship are as follows: "knots: 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 PEn kW: 544.5 760.8 1031.2 1364.9 1769.2 2252.2 2823.3 3490.7 A service allowance of 20 percent must be added. The propulsion factors based all thrust identity are: w =

0.200

t = 0.150

1]R

= 1.050

The main engine of the ship has a brake power of 5000 kW at 126 rpm, and is directly connected to the propeller, the shafting efficiency being 0.970. The depth of immersion of the propeller axis is 3.5 m. Based on vibration considerations, the propeller is to have four blades. The Burrill cavitation criterion for merchant ship

229

Propeller Design

propellers is to be used for determining blade area. The propeller is to belong to the methodical series for which the data are given in Tables 9.1 and 9.2. Design the , propeller. (a) I"" .

,~

Solution based on Optimum Efficiency Line data, Table9.1

V knots 10.0 11.0 12.0 13.0 14.0 15.0 16.0 17.0 P E = 1.2 PEn kW: 653.4 913.0 1237.4 1637.9 2123.0 2702:6 3388.0 4188.8 U'

. PB

h

=

0.200

t = 0.150

= =

5000kW

n

3.5m

Z=4

'TlR = 1.050,

= 126rpm = 2.1000s- 1

thrust identity 'TlS

=

0.970

!"

i' 1<

Optimum 'diameter and pitch ratio: PD

=

'TlH =

J

KQ

J5

1 - t = 1- 0.150 = 1.0625

1- 0.200

1- w

= \~4. = nD

V =

KQ

PB'TlS = 5000 X 0.970 = 4850 kW

= =

VA 1-w

=

Qo pn 2 D5

1- 0.200

=

PD'TlR 211"n pn 2 D5

= 1.2500 VA ms- 1 = 2.4300 VA =. 2'7rPD'TlR pn D5 3

n 5 D5 n 2 PD'TlR PD'TlR = 211" pVA5 211" pn 3 D5 VA 5

4

=

5.11085

[(KQ/J5)~r6

knots

I,

W4


230 PD 1/0 1/R 1/H

(~~Y 0.6500 0.7000 0.7500 0.8000 0.7237

D =

~I'1

= 4850 x 1/0 x 1.050 x 1.0625 = 5410.781/0 kW

From Optimum Efficiency

Line for As/Ao = 0.40

P J 1/0 D 0.6647 0.6220 0.5854 0.5533 0.6041

0.8840 0.8482 0.8184 0.7930 0.8336

0.6607 0.6453 0.6303 0.6156 0.6379

VA

V

PD 1).0 7)R 1/H

ms- 1

k

kW

7.2138 6.7985 6.4335 6.1097 6.6195

17.5296 16.5204 15.6333 14.8466 16.0855

3574.9

3491.6

3410.4

3330.9

3451.8 = Ps

I

6.6195 = 5.2179m 2.1 x 0.6041 ~

1.

(~~y 0.6500 0.7000 0.7500 0.8000 0.7259

D

=

From Optimum Efficiency

Line for AE/Ao = 0.55

P J 1/0 D 0.6671 0.6249 0.5889 0.5574 0.6057

6.6034 2.1 x 0.6057

0.8979 0.8625 0.8343 0.8104 0.8474

=

0.6562 0.6407 0.6254 0.6104 0.6325

'S4.

V

PD 1]0 1]R 7JH

ms- 1

k

kW

7.2138 6.7985 6.4335 6.1097 6.6034

17.5296 16.5204 15.6333 14.8466 16.0463

3550.6

3466.7

3383.9

3302.7

3422.4 = PE

5.1915m

.~

1 ~

'~

1.

(~~y 0.6500 0.7000

From Optimum Efficiency Line for AE/A o = 0.70 P J 170 D 0.6690 0.6276

0.9091 0.8753

0.65~9

0.6362

~

1 I i

VA

V

PD 7}0 'fiR 7}H

ms- 1

k

kW

7.2138 6.7985

17.5296 16.5204

3527.3

3442.3

Propeller Design

~

if

~

R

~

From Optimum Efficiency Line for AE/Ao = 0.70 P J 770

1

(~~ )4

D

0.7500 0.8000 0.7285

D=

231

0.5922 0.5613 0.6072

2.1

0.8483 0.8260. 0.8597

0.6208 0.6057 0.6273

VA

V

PD 1Jo 1JR 77H

ms- 1

k

kW

6.4335 6.1097 6.5878

15.6333 14.8466 16.0084

3359.0 3277.3 3394.2

= PE

6.5878 = 5.1664m X 0.6072

Cavitation:

I I

I

(jO.7R

= PA +1 pgh \1;2

7. P

= Tc

=

O.7R

PV

=

101.325 + 1.025 x 9.81 x 3.5 - 1.704 I

~ x 1.025 VO~7R

263.0505

VO~7R 0.0321 + 0.3886 (j5.7R - 0.1984 (j5.7R + 0.0501

ol7R

( Burrill cavitation criterion for merchant ship propellers.)

P

RT = E V T

1 v;2

T=

RT

RT

I-t

1 - 0.150

-- =

Ap

= 7. P O.7RTc = ~ x 1.025 VO~7R Tc = 0.5125 \1,2 O.7R Tc

A'g

=

Ap

1.067 - 0.229P/D

(Blade area required)

Ao = ~D2 4 The values of the various parameters for the different values of AE/Ao are pdt down in the following table and used to find the required value of A'g/Ao. by interpolation.


232 AE Ao Dm

0.40

0.55

0.70

0.5019

5.2179

5.1915

5.1664

5.2000

P D VAms- 1

0.8336

0.8474

0.8597

0.8430

6.6195

6.6034

6.5878

6.6086

Vk

16.0855

16.0463

16.0084

16.0589

PEkW

3451.2

3422.4

3394.2

3431.6

RTkN

417.09

414.62

412.18

415.41

TkN

490.70

487.79

484.92

488.72

v;2 m 2 s-2 O.7R

624.48·

618.41

612.66

620.36

(JO.7R

0.4212

0.4253

0.4294

0.4240

Tc

0.1643

0.1653

0.1663

0.1650

.2:... kNm- 2 Ap

52.5912

52.4018

52.2310

52.4648

Apm2

9.3305

9.3087

9.2841

9.3152

A Em 2

10.6500

10.6636

10.6698

10.6587

Aom 2

21.3836

21.1678

20.9636

21.2372

0.4980

0.5038

0.5090

0.5019

A~

-.li

Ao '

AE

= Ao

The values at which AE/Ao (first row in this table) and A'e/Ao (last row) are equal are obtained by interpolation. The particulars of the design propeller are therefore as follows:

D

= 5.200 m

P

D

=

0.8430

AE Ao

=

0.5019

V

=

16.059 k

The blade thickness fraction and the boss diameter ratio, which are fixed for a given methodical series of propellers, must be checked to complete the propeller design, and a propeller drawing prepared. It is also desirable to carry out pex'for mance estimates, i.e. to determine the variation of brake power and propeller rpm with speed for various ship operating conditions, as shown in Example 3.

Propeller Design

233

(b) Solution based on Regression Equations, Table 9.2

!'

r \ I

:t . ~

"

?1

The design of a propeller using regression equations for KT and K Q involves far too much calculation to be carried out by hand, even with the simplified equations of Table 9.2. A computer program for propeller design is more or less essential, and several propeller design programs are available. Different programs may approach the propeller design problem in different ways. A typical approach, summarised in Table 9.3, consists in determining the values of propeller diameter D, blade area ratio AE/A o , pitch ratio P/D and advance coefficient J by a process of systematic trial-and-error such that the highest speed V is obtained subject to the conditions: (a) the propeller thrust T is in balance with the total resistance RT at that sp~d, (b) the delivered power PD required by the propeller is equal to that delivered by the engine, and (c) the specified cavitation criterion (e.g. the Burrill criterion) is satisfied. /

A comptiter program based on this approach produces virtually the same result as that obtained by using the optimum efficiency line data: D

= 5.194m

P

_

D

= 0.8437

AE

-A 0

_

= 0.5030

V

= 16.052k

I

the small differences being due to round-off errors. Table 9.3 Outline of Propeller Design Program Start with the given input data

Choose a value of D

Choose a value of AE/ Ao

Choose a value of P / D Choose a value of J Calculate KT and KQ from equations Determine V for this value of J Calculate T from KT and RT from PE/V Adjust the value of J until T = (1 - t)RT From KQ determine PD Adjust the value of P / D until P D = PB 17s Adjust the value of AE/Ao until the cavitation criterion is satisfied Adjust the value of D until the highest speed is obtained.

----til

._._------_.

234


The propeller design parameters obtained by procedures similar to that used in the foregoing example are really optimum only for uniform flow. The flow behind a ship, particularly a si~gle screw ship, is far from uniform. Some designers therefore recommend that the optimum propeller diameter determined by such procedures be reduced by a small amount (2-5 per cent) for single ·screw ships with a consequent change in the other design parameters, viz. pitch ratio and 'blade area ratio. Since the mean circum ferential wake at any radius is greater at the inner radii in a single screw ship, a reduction in the propeller diameter results in an increase In the av erage wake fraction wand in the hull efficiencY1]H so that the propulsive efficiency 1]D = 1]0 1]R 1]H increases even if there is a small decrease in the open water efficiency 1]0. If the ~ri~tion in the propulsion factors w, t and 1]R with propeller diameter can be determined, it is possible to incorporate this variation in a propeller design;program to determine the exact optimum design parameters instead ofmaking 'a somewhat arbitrary reduction in the propeller diameter determined from the methodical series open water data. Example 3 Estimate the performance characteristics of a ship in a lightly loaded trial condition given the following data: Effective Power (naked hull):

V knots:

10.0

11.0

12.0

13.0

14.0

15.0

16.0

17.0

18.0

FEn kW:

419

574

765

996

1272

1597

1976

2414

2915

Trial·allowance: 10 percent Propulsion Factors (thrust identity): w = 0.180

t = 0.160

7'JR = 1.060

Propeller:

Z = 4

D == 5.200m

P/D = 0.8430

AE/A o = 0.5020

Open water characteristics from Table9.2 Engine:

Brake power

= 5000 kW

' rpm

= 126

Shafting efficiency = 0.970

Engine directly connected to the propeller.

y, ,

,

Propeller Design

235

P From Table 9.2, for D

= 0.8430

AE Ao

= 0.5020

KT = 0.4149 - 0.4413 J 10KQ KT J2

=

0.5296 - 0,5043 J

T

RT n2 D2 .1- t VA 2 = p D2 ( 1 - W )2 V2

PE

1

=

pn 2 D4

=

1 1 PE PE = 15.6544 Va 1.025 x 5.200 2 x (1 - 0.160) (1 - 0.180)2 V3

P D2 ( 1 - t)( 1 -

w)2

Va

0.4149 0.4413 KT Also, J2 = ] 2 - - J

'1 _ 0.4413 so that. } -

~ V(0.4413)2 + 4x 0.414~' x ~; 2 x 0.4149

= 0.5318 + [ 0.282827 +

K /J2

]0.5

0.~149!

,

from which J can be determiI,led, and henc~ KT' KQ and '70 using the equations for KT and K Q. . Then:

n =

VA = (1 - w) V = (1 - 0.180) V = 0.157692 V J D

PI)

=

.J D

J x 5.200

J

21l' pn 3 D 5 KQ· = 21l' x 1.025 x n 3 x 5.2005 KQ '7R 1.060

= 23100 n 3 K Q kW '7H

I

I 1!

L

=

1- 0.160 1-t 1- w = 1 - 0.180 = 1.0244

Alternatively,

PD = PE =

PE '70 '7R '7H

PD '75

=

= PD

0.970

PE '70 x 1.060 x 1.0244

1 PE

- =1.0859 '70

B- 1

236


The calculations of power and rpm at different speeds can then be carried out as shown in the following table (p.237). In this condition, at the rated propeller rpm of 126, the ship is estimated to have a speed of 17.49 k with the engine developing a brake power of 4166 kW, Le. about 80 percent of its normal rating. This type of information is often useful during the acceptance trials of a ship.

9.4

Design of Towing Duty Propellers

The design of propellers for tugs, towboats and trawlers is somewhat more complex than that of free running propellers because towing duty propellers are usually required to perform effidently in more than one operating con dition. A tug propeller, for example, may be required to produce a high static pull when the tug is attached to a bollard as well as to have a high speed when running free. The bollard pull condition and the free running condition represent the two extreme conditions of operation of a propeller in practice. A towing duty propeller may also be designed for maximum ef ficiency when towing at some intermediate speed between zero speed (static condition) and the free running (maximum) speed. Most tugs and trawlers today have propellers driven by diesel engines through reduction gearing with reversing arrangements. A diesel engine has the characteristic that the maximum torque that it can produce is nearly constant over a wide range of rpm, so that the maximum power available from the' engine varies almost linearly with rpm. A fixed pitch propeller, on the other hand, can absorb a given power at a given rpm only at a fixed speed of advance: the higher the speed the lower the power at a given rpm. A propeller designed to absorb the full power available from the engine at a particular speed of advance will therefore tend to overload the engine at lower speeds unless the rpm is reduced, while at higher speeds the engine will tend to run at higher rpms than its rated value unless the fuel supply is decreased because the power available from the engine is more than that absorbed by the propeller.

1

Various methods are available to overcome this problem. A multi-speed gearbox can be fitted between the engine and the propeller, so that low pro peller rpm: engine rpm ratios can be used at low speeds and higher values at

,t

. ,

\

f

I

. .J".""

~

r

. .

..

-,.";~",,',

..

'",:r'",''''

.-

,"-. '". c-,.-.

,HSL... ,•.

"·'~· ... ~",c"';"~"'Z,,=,!,"":"IF';"¢,(~q".~~~~ii6'

- Ja"J!3i"!

~;

c

r

'1::1 a 'tl

~ ~

Calculation of Performance Characteristics (Example 3)

t1 ~

V

PE = 1.1PEn

KT J2

§. J

KT

10KQ

n

7]0

5- 1

Pn

PH

rpm

kW

kW

k

ms- 1

kW

10.0

5.1440

460.9

0.2163

0.7000

0.1060

0.1766

0.6687

1.1588

69.53

634.8

654.4

11.0

5.6584

631.4

0.2226

0.6959

0.1078

0.1787

0.6681

1.2822

76.93

870.2

897.1

12.0

6.1728

841.5

0.2285

0.6921

0.1095

0.1806

0.6679

1.4064

84.39

1160.5

1196.4

13.0

6.6972

1095.6

0.2340

0.6887

0:1110

0.1823

0.6674

1.5312

91.87

1511.8

1558.6

14.0

7.2016

1399.2

0.2393

0.6854

0.1124

0:1840

0;6669

1.6569

99.41

1933.4

1993.2

15.0

7.7160

1756.7

0.2443

0.6824

0.1138

0.1855

0.6663

1.7830

106.98

2428.9

2504.0

16.0

8.2304

2173.6

0.2490

0.6796

0.115.0

0.1869

0.6655

1.9097

114.59

3006.9

3099.9

17.0

8.7448

2655.4

0.2537

0.6768

0.1162

0.1883

0.6647

2.0375

122.25

3675.2

3793.0

18.0

9.2592

3206.5

0.2580

0.6743

0.1173

0.1896

0.6639

2.1654

129.92

4447.0

4584.5

17.489

8.9963

2915.8

0.2558

0.6756

0.1167

0.1889

0.6642

2.1000

126.00

4041.1

4166.1

~

w

~


238

higher speeds. A diesel electric drive can be used instead of a geared diesel drive, the motor rpm being controlled to suit the operating conditions. In stead of using a fixed pitch propeller, a controllable pitch propeller may be used so that the pitch can be changed with changing speed. Controllable pitch propellers are discussed later (Chapter 12). However, these solutions to the problem of matching engine and propeller characteristics have the!r own disadvantages such as higher cost or lower efficiency, and most tugs and trawlers have diesel engines driving fixed pitch propellers through single speed reduction gearing. Sometimes, propellers in nozzles are used. The design of towing duty propellers therefore usually demands a compromise between the conflicting requirements of the static and the free running con ditions. Towing duty propellers are designed using methodical series data. Three or four-bladed propellers with aerofoil type blade sections (e.g. B-Series propellers) are normally used, although propellers with segmental sections (Gawn Series) are also used sometimes. The blade area ratio is determined from considerations of cavitation. The Burrill cavitation criterion for tug and trawler propellers may be used for this purpose. Alternatively, one may use the following empirical formula: I

AE _ K Ao nD3 [PA

PD + pg(h - 0.8R) - PV]

(9.2)

where:

K - 4.50 for open propellers

- 3.22 for propellers in nozzles

Once the number of blades and the blade area ratio have been selected, the thrust and torque coefficients become' functions of only the pitch ratio and the advance coefficient for a particulaI,: methodical series. In addition to these coefficients, one also requires data on the propulsion factors, which may be obtained from empirical formulas or from the data of similar vessels. The wake fraction, thrust deduction fraction and relative rotative efficiency vary in a somewhat complex manner with speed in a tug or trawler, but for design purposes; it is usual to assume that the wake fraction and relative rotative efficiency are constant while the thrust deduction fraction varies ,.

'

'

Propeller Design

239

linearly 'with speed from its value in the static condition (0.03-0.05 for open propellers and up to 0.12 for propellers in nozzles), to the value in the free running condition. Design problems of propellers for tugs and trawlers may be divided into three types: (i) The determination of the optimum propeller .rpm and the correspond ing delivered power to achieve both a specified bollard pull and a spec ified free running speed. (ii) The design of a propeller for a given propulsion plant. (iii) The estimation of the performance characteristics of a vessel with a given propeller and propulsion plant. The data required for a type (i) problem are the open' water character istics of the selected propeller series (KT and KQ as functions of J and P/D), the propulsion factors (w, t and17R) in both the static and the free running conditions, the required bollard pull free running , BP, the required . speed 11'1 and the corresponding effective ppwer PEl. The propeller diame ter D should be the maximum that can be accommodated in the propeller aperture keeping adequate clearances. The calculations are based on the assumption that in the bollard pull condition the torque is the maximum available from the engine with the propeller rpm being reduced, whereas in the free running condition the torque is less than the maximum available. with the propeller rurming at its full rpm. The procedure for determining the optimum propeller rpm and the corresponding delivered power is illustrated by the following example. Example 4 ,

,

The maximum propeller diameter: that can be accommodated in the aperture of a single screw tug is 3.10 m. The required blade area ratio (expanded) is estimated to be 0.500. The tug is required to have a bollard pull of 15 tonnes and a free running speed of 12 knots. In the bollard pull condition, the thrust deduction fraction is 0.050, and in the free running condition at 12 knots, the effective power is 285 kW, the wake fraction being 0.200 and the thrust deduction fraction 0.180. The relative rotative efficiency may be taken as 1.000. Determine the optimum propeller rpm and the corresponding delivered power. Use the open water data of Table 4.3.

240


D

== 3.lOm

Bollard pull, BP

= 15 tonnes

= 15 x 9.81 = 147.15 kN

Free running speed, VI = 12k = 6.1728ms- 1 WI

= 0.200

tl

= 0.180

f]R

PEl

== 1.000

=

285kW

to = 0.050

At the free running speed (subscript 1): KTI

J'f

PEl

1

= pD2 (1.,... td (1- w)2 VI3

== 1.025

X

3.102

X

1 (1- 0.180)(1 _ 0.200)2

2~

X

6.17283

= 0.23440

(1 -

O.~OO) X nl X

1.59298· 6.1728 = 3.10

The propeller revolution rate ni in the free running condition is obtained for different pitch ratios by finding the values of J at which KTfJ2 == 0.234~0. This may be done by finding the intersections of the KT - J lines for different pitch ratios with the curve KT = 0.23440 J2. .

P D J

0.5

0.7

0.8

0.9

1.0

1.1

1.2

0.4618 0.5287 0.5928 0.6554 0.7171 0.7781 0.8390 0.8973 0.0500 0.0655 0.0824 0.1007 0.1205 0.1419 0.1650 0.1887

KT ni

0.6

~l

s·

3.4492 3.0130 2.6872 2.4305 2.2214 2.0473 1.8987 1.7753

In the bollard pull condition (subscript 0), the bollard pull and the propeller torque are given by:

where KTO and KQO are the values at J

BP = (1 - 0.050) x 1.025 x

== O. Then:

n6 x 3.104 KTO

= 147.15 kN

Propeller Design

241

so that 1.63631

Kro Qo

=

qo 1.025 x n 2 x 3.10 5 K o 1.000

= 293.4488Kqo kNm

This is the maximum torque required by the propeller. The maximum revolutions are given by nl. Therefore, the engine must be capable of producing a delivered power P Dmax at nl revolutions per sec where:

The calculations for different pitch ratios are set out in the follQwing table:

P D

Kro

0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

0.2044 0.2517 0.2974 0.3415 0.3840 0.4250 0.4644 0.5022

lOKqo

n 02

no

0.1826 0.2455 0.3187 0.4021 0.4956 0.5994 0.7133 0.8374

8.0054 6.5010 5.5021 4.7915 4.2612 3.8501 3.5235 3.2583

2.8294 2.5497 2.3456· 2.1890 2.0643 1.9622 1.8771 1.8051

nl

Qo kNm

5- 1 I

·PDmax

S-1

3.4492 3.0130 2.6872 2.4305 2.2214 2.0473 1.8987 1.7753·

42.896 46.834 51.457 56.538 61.972 67.721 73.753 80.068

kW 929.6 886.6 868.8 863.4 865.0 871.1 879.9 893.1·

• Note that at PI D = 1.2 and above, nl is less than no. This violates the assumption that the torque is maximum in the bollard pull condition while the propeller revolution rate is maximum in the free running condition. The optimum value of pitch ratio is that at which

P D PDmax

PDmax'

has the lowest value:

= 0.827

no

= 2.15248- 1

= 863.1kW

nl

= 2.36968- 1 = 142.2rpm

Qo

= 57.969kNm

A tug or trawler propeller may be designed to absorb the full power avail able from the propulsion plant in the bollard pull condition, or in the free


242

running condition, or at a given speed for towing. The propeller can also be designed to produce a combination of high bollard pull and a high free running speed. ' If the propeller is to be designed for absorbing the full power in the bollard pull condition, and it is possible to select the optimum propeller revolution rate through a proper choice of the gear ratio between engine and propeller" the propeller diameter can be made the largest practicable. Then, the bollard pull is given by:

P ) BP = (1 ~ to) ( 411"2

l

J{TO

(9.3)

K 2/ 3 (71RPDO D) QO

in which all the quantities except KTO and KQo are known and thrust identity is assumed. The maximum bollard pull is obtained by choosing the' pitch ratio for which KTO/K;}03 is maximum, and the propeller revolution ,rate is then given by: ' PDO'f/R

(9.4)

Usually however, the power and revolutions must be regarded as fixed, i.e. a given engine and gearbox. In that case:

=

.1

1

(1 _ to)

,(--f!_) 5 KTO 1611"4 K 4/ 5

(9.5) vno QO and the pitch ratio must be selected to make KTO/ KiJo5 ma:l(:.imum, the BP

(71R PDO)

5

propeller diameter being given by:

D5 =

PDO 'f/R 211" P KQo

n8

(9.6)

If the propeller revolution rate no is too low, the diameter obtained may be greater than the maximum that can be fitted to the vessel with adequate clearances.

An advantage of designing propellers for the bollard pull condition is ,that there is little risk of overloading the engine. However, at all forward speeds

Propeller Design

243

the torque will be less than the maximum available from the engine, and the penalty in free running speed may be considerable. Another disadvaJ;ltage is that propellers designed for the bollard pull condition, Le. for [(TO![('tj03 or

[(To!lC'do5 maximum, usually have low pitch ratios so that when the'Ves'seris

running free, the blade sections at the outer radii may have negatiye a~glesof attack over a part of the revolution, and this may give rise to face .cavitation and blade vibration.

If a tug or trawler propeller is to be designed for the f~ee r~nnrrigibondi:.. tion, the propeller must be designed for the maximum effic~ency at If.Q/J5 constant at the speed for which:

PD1 1]O 1]R 1]H = PEl

:(9.7).

The procedure has been described in Section 9.3 (Example2), Designing a tug or trawler propeller for the free running condition has the disadvantage of very low bollard pull s~nce the propeller revolution rate hap, to be gre~tly reduced to keep the torque within permissibie limits. There is, thus,' also a danger of overloading the engine (exceeding the permissible torque) if the propeller revolutions are not reduced suitably during towing or trawling. If the propeller is to be designed for a specific 'towing speed, the diame ter and pitch ratio must be chosen for the maximum efficiency for [(Q!J5 constant at that speed. The available towrope power is then given by:

(9.8) A tug or trawler propeller may also be designed to give the optimum compromise between high bollard pull and high free running speed, since an increase in the one is associated with a decrease in the other. The de sign data required include the engine power, rpm, gear ratio and shafting efficiency so that the maximum delivered power PDmax and the maxirrium propeller revolution rate nl are known. The propeller diameter D is taken· to be the highest practicable when nl is equal to or less than the optimum determined as illustrated in Example 4; otherwise, the design calculations must be carried out for several diameters and the optimuPl diameter de termined. The bollard pull and the free running speed are determined for different pitch ratios for a given diameter, and the pitch ratio sel~cted to


244

give the optimum combination of ballard pull and free running speed. The .design procedure is illustrated by the following example. Example 5 A single screw tug is to have an engine of 900 kW brake power at 600 rpm driving the propeller through 4:1 reduction gearing, the shafting efficiency being 0.950, The wake fraction in the free running condition is 0.200, and the thrust deduction fraction may be assumed to vary linearly with speed with a value of 0.050 at zero speed and 0.180 at 12 knots. The relative rotative efficiency is 1.000. The effective power of the tug is as follows:

V knots:

a

2.0

4.'0

6.0

8.0

9.0

10.0

11.0

12.0

13.0

14.0

PE kW:

0

0.5

6.1

25.2

68.9

104.1

150.6

210.2

285.0

377.1

488.8

Design the propeller for both high bollard pull and high free running speed. Assume that the propeller diameter is 3.0 m, and its depth of immersion is 2;7~ m.

PB = 900kWat 600rpm

Gear ratio

= 0.200

to = 0.050

D - 3.0m

h = 2.75m

w

PDmax

Qmax

=

= PB 1]8

=

Qma-'C and respectively.

PDmax 211"n max

n max

900 x 0.950

=

855 211" x 2.5

= 4: 1

t1

= 0.180

-

855kW

at VI

'7]8

= 0.950

= 12.0 k n max

7]R

' 600 4

= 1.000

=

150 rpm

=

2.55- 1

= 54.431 kN m

are the limiting values of propeller torque and revolution rate

Preliminary estimation of blade area ratio: AE _

Ao -

45

PDmax TJR

. n max D3[PA+pg(h-0.8R)-pv)

_ 45 . 2.5

= 0.4948

X

855 x 1.000 3.0 3 [101.325 + 1.025 X 9.81 (2.75 - 0.8

X

1.5) - 1.704)

Propeller Design

245

One may therefore use the data of Table 4.3 since it is for a blade area ratio of· 0.500. It is necessary to plot the data to obtain KT - J and KQ - J curves for different pitch ratios.

In the bollard pull condition: KQo

_

-

Qmax 7JR pn~D5

so that

=

54.431 X 1.000 KQo x 1.025 x 3.05

0.2185 -2 = --s K QO

BP

PD~

=

(l-to)KTOpn~D4

-

78.8738 KTO n~ kN

= 21r P n5 D5 KQo

=

=

(1-0.050)KTOX1.025xn~x3.04

21r x 1.025 x n5 x 3.05 KQo

7JR

1.000

= 1564.9844 n~ K Qo kW

The calculation of the bollard pull and the corresponding revolution rate and delivered power for different pitch ratios is carried out in the following table:

P

0.5

0.6

0.7

0.8

0.9

1.0

0.2044

0.2517

0.2974

0.3415

0.3840

0.4250

10KQo

0.1826

0.2455

0.3187

0.4025

0.4956

0.5994

1

2.5000

2.5000

2.5000

2.3313

2.0999

1.9094

BPkN

100.76

124.08

146.61

146.39

133.55

122.21

PDokW

446.51

600.32

779.31

797.33

718.19

653.01

D KTO

nos-

=

(no nma.x and Qo P D > 0.7386.)

<

P

QmaJ<

for D

<

0.7386; no

<

n max

and Qo = Qmsx for


246 .In the free running condition:

(1 - 0.200) VI = 0.1067 VI 2.500 x 3.000

RT1 =

T1

p

ni D4

= 1.025

X

=

1- t1 p ni D4

1

PEl

(1 -

td VI

1 PEl = 0.001927 PEl· 2.500 2 x 3.04 (1 - t1 ) VI ( 1 - h ) VI

= = 24452.88 KQ1

The values of J~ and

= p ni1D4

KT1

271' x 1.025

X

2.500 3 x 3.0 5 KQ1 1.000

(PDl ':f PDmax )

are calculated for the different given speeds:

{ knots -1 ms

8.0

9.0

10.0

11.0

12.0

13.0

14.0

4.1152

4.6296

5.1440

5.6584

6.1728

7.2016

kW

68.9

104.1

150.6

210.2

285.0

6.6872' 377.1 ,.

t1

0.1367

0.1475

0.1583

0.1692

0.1800

0.1908

0.2017

J1

0.4390

0.4938

0.5487

0.6036

0.6584

0.7133

0.7682

0.0374

0.0508

0.0670

0.0862

0.1085

0.1343

0.1638

VI

PEl

KT1

488.8

\

Plotting this KT"l - J 1 curve on the K T - J diagram derived from Table 4.3, one may determine the values of J, KT and K Q at which the KT1 - J 1 curve intersects the KT - J curves for the different pitch ratios. The calculations are carried out as fonows:

p. 0.5

0.6

0.7

0.8

0.9

1.0

J1

0.4725

0.5349

0.5931

0.6482

0.7010

0.7515

KT1

0.0452

0.0627

0.0823

0.1041

0.1282

0.1545

D

0.0647

0.0901

0.1228

0.1629

0.2106

0.2667

U1S- 1

4.4297

5.0147

5.5603

6.0769

6.5718

7.0453

k

8.6114

9.7486

10.8093

11.8135

12.7758

13.6961

158.09

220.39

300.33

398.24

514.90

652.2i

101(Q1

F1 { PDlkW

Propeller Design

247

Finally, one may plot the ballard pull and the free running speed as a function of the pitch ratio, and select that pitch ratio which best meets the design requirements. One could, for example, choose to maximise a function such as:

F=aBP+bV1 where a and b represent the relative weights of the ballard pull and the free running speed. More simply, one could merely select a pitch ratio that gives an acceptable combination of ballard pull and free running speed. Thus, if one chooses a pitch ratio of 0.850, the following values are obtained:

P

= 139.77kN

D

=

0.850 :

BP

no

=

2.20945- 1

= 132.6 rpm

VI nl

= 12.300k =

2.5000s- 1

,

PDO = 755.48kW

=

150 rpm

PDl '= 454.19kW

To complete the design of the propeller, it is necessary to check that its strength, is adequate. '

The determination of the performance characteristics of a tug or trawler propeller involves calculating the delivered power, propeller revolution rate, and the maximum towrope pull or towrope power as a function of the speed of the vessel. There is a particular speed Vm at which the propeller absorbs the maximum torque available from the engine at the maximum revolution rate, i.e. at the speed Vm , the delivered power is equal to PDmax' Below this speed, the propeller revolution rate must be reduced to limit the torque to its maximum value. At speeds above Vm , even at the maximum propeller revolution rate, the torque and hence the delivered power remain below their maximum values. The calculations required to determine the performance characteristics of a towing duty propeller are illustrated by the following example. Example 6 Determine the performance characteristics of the tug propeller of Example 5. D

3.0m

P = 0.850 D

A E = 0.500

Ao

Z

=

4

248 w


= 0.200

==

7]R

to

1.000

= 0.050

h

Gear ratio == 4 : 1

PH = 900 kW at 600 rpm

==

== PE TIs 600

4

=

.

=

= 0.180

at 12 knots

T/s == 0.950

900 x 0.950 == 855 kW

150rpm

= 2.58

1

The open water characteristics of the propeller, interpolated from Table 4.3, are,:

J

o

KT

0.3630 0.3355 0.3048 0.2710 0.2340 0.1938 0.1504 0.1039 0.0542 0.0014

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

lOKQ : 0.4476 0.4211 0.3904 0.3553 0.3160 0.2724 0.2244 0.1722 0.1157 0.b550

At the speed Vm

:

PD

=

PDmaJ<

== 855 kW

n == n max = 2.5 s-l

=

271"

X

855 x LOOO 1.025 X 2.5 3 X 3.0 5

== 0.0350

For this value of K Q . J = 0.3151

V

m

=

JnD (1- w)

==

0.3151 X 2.5 X 3 = 2.9540ms- 1 = 5.7426 k 1 - 0.200

\.

At speeds below l'm, Q

=

Qma.'C

=

=

211"

855 X 2.5

= 54.4310 kN m

54.4310 X 1.000 1.025 X n 2 X 3.0 5

J ==

VA

nD

=

0.2185

(l-w)l! = (1- 0.200) V = 0.2667 l! nD n X 3.0 n

from which n for a particular V can be determined.

,

Propeller Design

249

The towrope pull T P and the towrope power PThw are then given by:

=

TP

(1 - t) T - RT where RT =

PE V

PTow = TP x V

At speeds above lIm'

=

J

and

n = n max = 2.5 s-l

(1 - w) l' __ (1 - 0.200) l' nD 2.5 x 3.0

== 0.1067l' ,

from which K T and K Q may be determined and the various quantities calculated as before. The calculations may be carried out as shown in the following table.

o

2.0

o o 0.0500

4.0 '

5.7426

6.0

8.0

1.0288

2.0576

2.9540

3.0864

4.1152

0.5

6.1

21.6

25.2

68.9

0.0717

0.0933

0.1122

0.1150

0.1367

0.4860

2.9646

7.3121

8.1648

16.7428

0

0.1196

0.2290

0.3151

0.3292

0.4390

KT

0.3630

0.3297

0.2953

0.2656

0.2605,

0.2187

10KQ

0.4476

0.4154

0.3806

0.3496

0.3443

0.2995

n{

2.2094

2.2934

2.3959

2.5000

2.5000

2.5000

rpm:

132.57

137.60

143.75

150.00

150.00

150.00

kN:

147.12

143.98

140.74

137.82

135.16

113.47

kNm:

54.431

54.431

54.431

54.431

53.598

46.624

kN :

139.77

133.17

124.64

115.04

111.45

81.22

137.00

256.47

339.84

343.98

334.22

{ l'

k: ms- 1

PE

:

kW:

t

RT

kN:

J

T

Q TP

PTow kW:


250 PD

kW:

755.48

784.34

819.19

855.00

841.91

732.63

PB

kW:

795.24

825.62

862.31

900.00

886.22

771.19

9.0

10.0

11.0

12.0

12.300

13.0

14.0

4.6296

5.1440

5.6584

6.1728

6.3271

6.6872

7.2016

104.1

150.6

210.2

285.0

310.7

377.1

488.8

0.1475

0.1583

0.1692

0.1800

0.1832

0.1908

0.2017

22.4857

29.2768. 37.1483

46.1703

49.1062

56.3913

67.8738

J

0.4938

0.5487

0.6036

0.6584

0.6749

0.7133

0.7682

KT

0.1963

0.1731

0.1488

0.1236

0.1159

0.0975

0.0704

10KQ

0.2752

0.2496

0.2226

0.1945

0.1857

0.1821

0.1673

n{

2.5000

2.5000

2.5000

2.5000

2.5000

2.5000

2.5000

rpm:

150.00

150.00

150.00

150.00

150.00

150.00

150.00

T

kN:

101.89

89.80

77.23

64.16

60.14

50.59

36.53

Q

kN m:

42.841

38.856

34.653

30.278

28.908

28.348

26.044

kN:

64.38

46.31

27.01

6.44

PTow

kW:

298.03

238.21

152.86

39.76

o o

PD

kW:

672.94

610.34

544.32

475.61

454.19

454.09

409.10

kW:

708.36

642.46

572.97

500.64

478.09

477.99

430.63

PE

kW:

t

RT

kN:

TP

\.

PE

The values of towrope pull and delivered power at zero speed tally with the values of bollard pull and delivered power obtained in the previous example. Similarly, the towrope pull is zero at the free running speed whose value and the corresponding delivered power tally with the values in the previous example.

9.5

Propeller Design using Circulation Theory

The circulation theory of propellers, discussed in Section 3.5, may be used to design a propeller in detail to obtain a prescribed distribution of loading

251

Propeller Design

along the radius, a pitch distribution matching the mean circumferential wake at each radius and blade section shapes that fulfil desired cavitation and strength requirements. Circulation theory methods are used for the design of propellers that are likely to have cavitation problems and work in very non-uniform velocity fields. In such cases, these theoretical design methods offer significant advantages with respect to efficiency and cavitation over methods based on experimental propeller methodical series data. Although there have been considerable advances in propeller theory and the design methods based on it, particularly since the advent qf computers so that many empirical corrections required in the earlier versions of t~e theory have been eliminated, it is still necessary to apply some corrections and to carry out experiments in a cavitation tunnel to confirm the correctness of a propeller design based on theory. As with design methods using I?:lethodical series data, several methods exist for designing propellers using the circulation theory. All these methods are fundamentally the same, but have differences only in detail. One such method is considered here. It is assumed that the following quantitiesl are known, possibly through a preliminary design calculation using methodical series data: the ship speed V and the corresponding propeller thrust T, the propeller diameter D, the number of blades Z, an estimated expanded blade area ratio AE/Ao, the depth of immersion h of the shaft axis and the propeller revolution rate n. For a "wake adapted" propeller, the mean effective circumferential wake fraction W (x) as a function of the non-dimensional radius x= r / R must also be known. This may be determined by a wake survey as described in Section 8.5, but must then be corrected for the difference between effective and nominal wake as follows: 1-w

1 - w(x)

=

1_

W

nom

[1- wnom(x) 1

(9.9)

where w is the effective wake fraction determined through a self-propulsion test using thrust identity, wnom(x) is the mean nominal circumferential wake fraction at radius x determined through a wake survey, and wnom is the volumetric mean nominal wake fraction given by:

~

{l[l-w nom (x)]xdx (9.10) 1-xb JXb Xb being the non-dimensional boss radius. The mean velocity of advance of ' the propeller is therefore: 1-wnom =

-----------~---,--, .._


252 ~

= (l-w)V

(9.11)

The next step is to determine the thrust loading coefficient: (9.12)

where Ao

= 1i D2/4 is the disc area of the propeller, and the advance ratio: (9.13)

One may then estimate the ideal thrust loading coefficient (i.e. for an inviscid fluid): CTLi

= .( 1.02 to 1.06) CTL

(9.14)

Alternatively, one may make an initial estimate of the average drag-lift ratio, tan" for the whole propeller, using the following relation: I tan, =

°z41~

-

0.02

(9.15)

and then obtain: CTLi

\

= 1- CTL 2A tan,

(9.16)

Next, it is necessary to estimate the ideal efficiency 7]i of the propeller. This is usually done with the help of the Kramer diagram, Fig. 9.2, which gives 7]1 as a function of CTLi, A and Z. However, one may also select any suitable value and arrive at the correct 7]i by the iterative process described in the following. This allows the hydrodynamic pitch angles (see Fig. 3.11) to be determined at the various radii x:

tan,6 =

[l-w(x)]V = [l-w(x)]V 27rnr 7rnDx

1 tan,6r = - tanl3 [1 7]i

~~~x)] ~

(9.17)

, (9.18)

253

Propeller Design

Figure 9.2 : Kramer Diagram.

or

tan/31

=

~i tan/3 [l~~~X)]~

(9.19)

Equations (9.18) and (9.19) are the optimum criteria for wake adapted propellers proposed by Lerbs (1952) and by van Manen (1955) respectively, and are based on making the product of thl} ideal efficiency and the hull efficiency constant over the radius.


254

Sometimes, it is necessary to decrease the loading on the blades near the tip to reduce the risk of cavitation. In such a case, the values of tanf3[ may be reduced at the outer radii and increased at the inner radii by multiplying the values of tanf3[ obtained from Eqns (9.18) or (9.19) by an appropriate pitch distribution factor for each radius. A typical radial distribution of hydrodynamic pitch to give reduced loading over the outer radii has been suggested by O'Brien (1962) and is given in Table 9.4, which shows the ratio of the tangent of the hydrodynamic pitch angle at a non-dimensional radius x to that at 0.7R. The loading on the blades may also be reduced at the inner radii to minimise hub vortex cavitation. Table 9.4 Radial Distribution of H,ydrodynamic Pitch Angle for Reduced Thrust Loading at Outer Radii

tanf3[(x)

0.2 :::; x :::; 0.4 :

( ) = 1.11 - 0.1x tanf3[ 0.7

0.4:::; x:::; 0.6:

tanf3[(x) tanf3[(0.7) -

1.03 + 0.3x 0.5x

0.6 :::; x :::; 1.0 :

tanf3[(x) {3 ( ) tan [0.7

1.21 - 0.3x

2

After determining the hydrodynamic pitch angles at the different radii, one may calculate the local advance ratios including the induced velocity components: >..[ = x tanf3[ (9.20) This enables one to obtain at each radius the Goldstein factors K. which are given as functions of 1/>"[, the non-dimensional radius x and the number of blades Z, Appendix 6. One may now calculate the radial distribution of the ideal thrust loading coefficient from Eqn. (3.39) after including the Goldstein factor: dGTLi

dx

-

!Ut VR

8xK. VA ~ cosf3[

~;{

(9.21)

~fi

!~

where, from Eqns. (3.28):

~, '."."'j.

;;

~ ~ .'

-'"

~.

,

.

Propeller Design

1

2

Ut

"V4

=

255

sin {3I sin( (3[ - (3) sin{3

tan {3I ( tan{3[ tan (3) tan{3 (1 + tan 2 (3I)

(9.22) cos ( (3I - (3) VR = sin{3 VA Eqn. (9.21) is sometimes written in terms of a non-dimensional circulation defined by:

G=

(9.23)

so that:

dCTLi =

[_1__ !Ut]

4Z G

dx

tan{3

VA

(9.24)

j

Integration then yields a value of the ideal thrust loading coefficient:

1 1

· C TLi =

Xb

\

dCTLi d -d- X

x

(9.25)

This "alue of CTLi must be equal to the initial value of CTLi calculated from Eqn. (9.1-1) or Eqn. (9.16). If the two values of CTLi are not in agreement, the value of 1]i must be altered and the calculation repeated until the initial value of CTLi from Eqns. (9.14) or (9.16) and the final value obtained from Eqn. (9.25) are in sufficiently close agreement. Eckhardt and Morgan (1955) suggest that the number of iterations to bring this about can be reduced by using the following empirical relation: 1]ik

1]i(k+l) =

1 + CTLiO - CTLik 5 CTLiO

(9.26)

where 1]ik and 1]i(k+l) are the values of 7]i for the kth and (k + l)th iterations, CTLiO is the desired value of CTLi, and CTLik ~he value obtained ,in the kth iteration. However, see the procedure adopted in Example 7.


256

Once the radial distribution of hydrodynamic pitch angle for a specified ideal thrust loading co~fficient has been determined, the values of the product CL cj D at various radii can be obtained, since as shown in Section 3.5:

(9.27) The remaining design process consists in determining the shape of the blade sections at the different radii and their pitch angles. The lift coeffi cient CL depends upon the type of aerofoil section, and its camber ratio, thickness-chord ratio and the angle of attack. It is necessary to choose these geometrical patameters such that the desired values of CL cj D are obtained at the different radii subject to the requirements of minimum risk of cav itation, adequate blade strength and minimum drag. Various alternative procedures are available for designing the blade sections. The types of aerofoil section generally used in propellers designed using the circulation theory are the Karman-Trefftz section and the NACA sections with a 0.8 and a 1.0 mean lines and NACA-16 and NACA-66' thick ness distributions. (XACA stands for the National Advisory Committee for Aeronautics, USA, now the National Aeronautics and Space Administration, NASA). The basic Karman-Trefftz section is built up of two circular arcs, but it is usual to use the circular arc mean line of this section with other thickness distributions such as the NACA-16 and NACA-66 distributions. The NACA a = 0.8 and a = 1.0 mean lines indicate the fraction of the' chord over which the suction pressure on the back of the section is constant at the "'ideal" angle of attack. The a == 0.8 mean line has been found to give exc~llent results in practice and is widely used along with the NACA-66 (modified) thickness distribution in propeller design. The geometrical details of the various types of sections used in propellers designed by the circulation theory are given in Appendix 2.

=

I ff

=

\

The hydrodynamic characteristics of such sections are generally available in the form of diagrams giving CLj(tjC) for cavitation inception as a function of the minimum pressure coefficient -Cpmin for various values of the thickness chord ratio tje and the camber ratio f je for "shock free entry" (or ideal angle of attack at which the suction pressure variation does not have a sharp peak). The hydrodynamic characteristics of section shapes used in propellers, are also given in the form of "bucket diagrams" which indicate the range of

f

I

257

Propeller Design

angles of attack for which the section will not cavitate for a given - Cpmin (see Fig. 6.7). Data for some sections used in propeller design are given in . Appendix 7.

~!

~, ~.

f

I I,

Although the NACA sections have been widely used in propeller design, better methods for designing aerofoi1s for a prescribed pressure distribution ha~'e now been developed. In blade sections designed by this new procedure (Eppler-Shen sections), the leading edge is unloaded and the loading shifted as far towards the trailing edge as possible without causing flow separation. This causes an increase in the cavitation free zone of the angle of attack and increases the cavitation inception speed by as much as 2-3 knots. After having selected the type of section to be used, one may use its hydrodynamic characteristics to determine the blade section geometry, i.e. the chord c, the thickness t and the ca~ber f at the various radii r. The value of the minimum pressure coefficient is calculated at each radius from:

-c

_

Pmin

\

-

PA

+ Pg ( h 1

x R) - PV

V2

(9.28)

2P R

and a suitable variation of blade thickness t with radius assumed. The values of tic and fie are then determined so that the required value of C[, cl D is obtained at each radius without exceeding the limiting value of CL/(tlc) for cavitation inception at the calculated - Cpmin ' The value of -Cpmln is sometimes reduced by up to 20 percent to allow for the non-uniform flow field. This procedure for selecting tic and fie often leads to an "unfair" variation of f and c with radius r, and it is necessary to fair these values to obtain a smooth variation. It is recommended that tic should not exceed 0.22. After having determined the fair values of c and t at each radius, the value of CL can be calculated from Eqn. (9.27). Alternatively, one may first select a suitable blade thickness distribution satisfying a given strength criterion and a suitable blade width distribution, i.e. the variation of c with r. A linear variation of blade thickness is often adopted: t(x) = (1 _ x) to + x tl (9.29)

D

D

D

where t(x) is the blade thickness at the non-dimensional radius x, tolD the

blade thickness fraction and iI the tip thickness; a common value for tIi D is

0.003. Two standard blade width distributions 'used in propeller design are

258


one used in the Troost B Series (described in Section 4.6) and one proposed by Morgan, Silovic and Denny (1968). These chord distributions are given by: C(x) Cl(X) AE (9.30) D = -Z Ao where the values of Cl(X) are given in Table 9.5. Table 9.5 Blade Chord Distribution

C(x) D

Cl(X) AE

=

--z- Ao

Values of Cl(X) "

x

B Series

Morgan

0.20 0.30 0.40 0.50 0.60 0.70 ,0.80 0.90 0.95 1.00

1.662 1.882 2.050 2.152 2.187 2.144 1.970 1.582 1.274 0.000'

1.6338 1.8082 1.9648 2.0967 2.1926 2.2320 2.1719 1.8931 1.5362 0.0000

..

$.,;

A suitable distribution of skew to define the shape of the expanded blade , outline may also be selected at this stage. The distribution of skew must be selected to minimise unsteady propeller forces due to a non-uniform wake. Unsteady propeller forces are considered in Chapter 11. A typical distribu tion of skew given by ::-dorgan, Silovic and Denny is as follows:

skew(x) R

= Rs

l R 2s

(

x 0.2 )2 J0.5

Rs =

,~,

'---

.\ 5 i

!(

where:

~"'.~

(9.31)

0.32 skew(1)

skew(1) + ---'-" 2

259

Propeller Design and

skew(l) =

Os 57.3coslh(1)

Os being the skew angle in degrees and (3[(1) the hydrodynamic pitch angle at the blade tip. Once the blade chord distribution has been selected, the value of GL can be calculated at each radius from Eqn. (9.27). It is usual to assume that at the ideal angle of attack the drag coefficient CD = 0.008 and this allows the lift-drag ratio at each radius to be calculated: C:J

=

tan'Y = C

L

0.008

CL

(9.32)

One may then determine the thrust-loading coefficient for viscous flow:

CT L =

dCTL'

-d . 1 ( 1 - tan Ih tan 'Y) dx

1

1.0

"

(9.33)

X

Xb

and this should agree sufficiently well with the initial value at the start of the design calculation obtained from Eqn. (9.12). If the value of GTL obtained from Eqn. (9.33) differs widely from the value obtained from Eqn. (9.12), the value of CTL must be altered and the entire calculation starting with an initial estimate of 1]i repeated until a satisfactory agreement is obtained between the initial and final values of CTL. After this, one may determine the power coefficient:

Cp -

PD ~pAo

'A3

_ -

1

1

Xb

dCTLi tan{3] + tan'Y dx dx tan{3

(9.34)

The delivered power PD obtained from this value of Cp must match the design delivered power of the ship propulsion plant; otherwise, the ship speed and the corresponding propeller thrust must be changed and the calculations repeated until the calculated Gp matches the delivered power available. The efficiency of the design propeller in the behind condition is obtained as: CTL (9.35) 1]8 = - Cp

B8Sic Ship Propulsion

260

After these iterative processes have converged to satisfactory values of GTL and Cp, one may proceed with the design of the blade sections. The camber ratio fie and the ideal angle of attack ai are determined at each radius from the corresponding values of CL: (9.36)

where k 1 and k2 depend upon the type of mean line and are given in Table 9.6.

Table 9.6

Camber Ratio, Ideal Angle of Attack

and Viscosity Correction Factor (J-L)

-fc =

k1CL

Mean Line NACA a:::: 0.8 NACA a :::: 0.8 (modified) NACA a =1.0 Circular Arc

a'?~

=

k 2 CL

k1

k2

I"

0.06790 0.06651 0.06515 0.07958

1.54 1.40 0 0

1.05 1.00 0)4 0.80

The value of aj should lie within the cavitation free zone as indicated in the "bucket diagram" illustrated in Fig. 6.7. The maximum and minimum values of the angle of attack for cavitation free operation with a given -Cpmin , tic and f / e for blade sections with different types of mean line and thickness distributions may be determined from the data in Appendix 7. The values of fie and ai obtained from Table9.6 are correct if the lift coefficient is due to a lifting line in inviscid flow. Since the propeller blades are like lifting surfaces, have a finite thickness and operate in a viscous flow, it is necessary to correct the values of camber ratio and angle of attack to account for lifting surface, thickness and viscosity effects. Experience has shown that for the NACA a:::: 0.8 mean line, the correction for viscosity is very small. Blade sections with the NACA a :::: 0.8 mean line and NACA-66 (modified) thickness distribution are therefore often preferred for the design of propellers. For other types of sections, a small incre'ase in

Propeller Design

261

the camber ratio or the angle of attack or both is necessary to obtain the required lift coefficient. A method for making this correction suggested by van Manen (1957) is as follows: Increase in angle of attack,

b.Ci

o

_ 0.7 1 - J1 k J1

Increase in camber ratio, b. £ C

£ x 57.3

C

0.35 1 - J1 k J1

£

(9.37)

C

where J1 is the viscous correction factor given in Table 9.6 and k is the Ludwieg-Ginzel curvature correction factor which is a function of the number of blades, the blade area ratio, the non-dimensional radius and the hydro dynamic pitch angle. Lifting surface corrections may be made using the factors due to Morgan, Silovic and Denny (1968):

£C corrected Cii o

Cit

= ICc

£C

corrected = k a .

Cii

(9.38)

7 k"t to D

= 5 .3

where k c and k a are lifting surface correction factors for camber ratio and angle of attack respectively, and Cit is a correction to the angle of attack to account for the finite thickness of the blade, kt being the corresponding factor and tolD the blade thickness fraction. These correction factors, which are given in Appendix8, have been derived under certain restricted conditions (such as constant hydrodynamic pitch over the radius, and NACA a = 0.8 mean line and NACA-66 thickness distribution), and it is not strictly correct to regard these factors as applicable in all cases. The pitch angle at each radius is then obtained: cp = f31

+ Cii + Cit

(9.39)

and the pitch ratio at each radius becomes:

P(x) D

=

7f X

tan'cp

(9.40)


262

This concludes the design of the propeller using lifting line theory with lifting surface corrections. The following example illustrates the procedure for such a design. ' Example 7 Design a propeller for a single screw ship using the circulation theory. The ship is to have a speed of 24.5 knots at which its effective power is 16500 kW. The propeller is to have six blades, a diameter of 7.0 m, an expanded blade area ratio of 0.775 and a blade thickness fraction of 0.060. The skew angle is to be 15 degrees. The propeller shaft is 6.50 m below the load water line of the ship. The propeller is to run at 108 rpm. The effective wake fraction is 0.220 and the thrust deduction fraction 0.160. The radial variation of nominal wake is as follows:

x = r/R: wnom(x) :

0.2 0.418

0.3 0.350

0.4 0.302

0.,5 0;269

0.6 0.246

0.7 0.227

0.8 0.215

0.9 0.204

1.0 0.192

The propeller blades are to be unloaded at the tip in accordance with Table 9.4. The blade chord distribution is to be in accordance with Table9.5 (Morgan) and the blade sections are to have the NACA a = 0.8 mean line and the NACA-66 (modified) thickness distribution.

v = 24.5k = 24.5 x 0.5144 = 12.6028ms- 1 , n 108 rpm = 1.8s- 1 FE = 16500kW D

= 7.0m

Z = 6

As 0.775 = Ao

w

=

0.220

to = 0.060 D

t =

h = 6.5m

Blade chord distribution according to Morgan Radial load distribution according to Table 9.4 Blade sections:

NACA a = 0.8 mean line

NACA-66 (modified) thichkness distribution

Distribution of. Effective Wake This is calculated using Simpson's Rule as follows:

0.160

Propeller Design x

WnoJ'n

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.418 0.350 0.302 0.269 0.246 0.227 0.215 0.204 0.192

263 1-

= 1 - w(x) =

=

0.1164 0.1950 0.2792 0.3655 0.4524 0.5411 0.6280 0.7164 0.8080

0.582 0.650 0.698 0.731 0.754 0.773 0.785 0.796 0.808

~ x~

1 - Wnom . = 1

8M /(1 - Wnom )

x(l - w norn )

W nom

1:.

1 4 2 4 2 4 2 4 1

0.1164 0.7800 0.5584 1.4620 0.9048 2.1644 1.2560 2.8656 0.8080 10.9156

0

1-

(

W nom )

x dx

=!

1 - w(x)

1 _20.2 2

X

0.5989 0.6689 0.7183 0.7522 0.7759 0.7954 0.8078 0.8191 0.8315

~ x 0.1 x 10.9156

0.7580 1

1-ill -

-

W no7n

(1-

1.0290 ( 1 -

W nom )

=

1':"" 0.220 . 0.7580 (1-

) W nom

W norn )

Thrust loading coefficient and ideal efficiency: \~

=

Rr

=

T

=

--

Rr 1-t

=

1309.23 1 - 0.160

Ao

=

'!!- D 2

=

-

GrL

=

,\

=

I

I i

L 1

~.

-

=

(1- ill)V

PE V

=

4

16500 12.6028

7f

4

T V 2 'ipAO A 1

VA

7fnD

(1 - 0.220) x 12.6028

7f

=

9.8302ms- 1

1309.23 kN

X

7.0 2

=

1

= ::::

1558.61 m 2 38.4845 m 2 1558.61

'2

x 1.025 x 38.4845 x 9.8302 2

9.8302 x 1.8 x 7.0

=

0.2483

=

0.8178


264 tan'1 =

0.4 x 0.775 °Z4 ~; - 0.02 = 6"" GTL

1 - 2>' tan '1

= 0.03167

0.02

0.8178 = 0.8309 1 - 2 x 0.2483 x 0.03167

=

=

From the Kramer diagram for Z =6, >. 0.2483 and efficiency: 7]i = 0.800 Hydrodynamic pitch angles-first iteration with

tan{J

=

[l-w(x)]V

=

1fnDx

1]i

GTLi

= 0.83tl9, the ideal

= 0.800

= 0.31838 1 -

[1 - w(x) I x 12.6028 1f x 1.8 x 7.0 x x

-

w(x)

x

Optimum tan {JI __ tan {J [ 1 - w ] 7]i 1 - w(x)

~

..

(Lerbs)

tan{JI = Optimum tan{JI x tip unloading factor (tuf) .

=

1

tan{J [ 1 - 'Iii ] 7]i 1 - w(x)

2

X

tuf

The calculations are carried out in the following table: x

1- w(x)

tan{J

{J0

O.~

0.5989 0.6689 0.7183 0.7522 0.7759 0.7954 0.8078 0.8191 0.8315

0.9534 0.7099 0.5717 0.4790 0.4117 0.3618 0.3215 0.2898 0.2647

43.6331 35.3703 29.7580 25.5931 22.3779 19.8887 17.8218 16.1596 14.8280

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

[

I-til

1- w(x) 1.1412 1.0799 1.0421 1.0183 1.0026 0.9903 0.9826 0.9758 0.9685

]~

tuf

tan{JI

{JI

{JI - {J0

1.090 1.080 1.070 1.055 1.030 1.000 0.970 0.940 0.910

1.4824 1.0349 0.7968 0.6432 0.5315 0.4478 0.3830 0.3322 0.2917

55.9979 45.9818 38.5478 32.7491 27.9901 24.1235 20.9586 18.3788 16.2598

12.3648 10.6115 8.7919 7.1564 5.6122 4.2348 3.1368 2.2192 1.4318

(The \-alues in this table and other tables have been rounded off to four decimals, and this may result in minor discrepancies.)

I

Propeller Design >'1

265

= x tan fJI

/'\, from Appendix 6 fJI = costsinfJ

sinfJI sin(fJI - fJ) sinfJ dCTLi

~

fJ)

-~--=-.;....:.

~Ut VR = 8x/'\,- - cosfJI VA VA

CTLi is calculated using Simpson's Rule as follows:

x

>'1

/'\,

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.2965 0.3105 0.3187 0.3216 0.3189 0.3135 0.3064 0.2990 0.2917

1.0529 1.0078 0.9913 0.9855 0.9783 0.9565 0.9035 0.7563 0

~Ut VA 0.2573 0.2288 0.1919 0.1560 0.1206 0.0887 0.0640 0.0439 0.0273

GTLi =

31 X 0.1 x 22.8148

VR dOTLi ~ VA 1.4156 0.3431 1.6980 ;0.6530 1.9911 :.·0.9479 2.2969 1.1880 2.6141 :' 1.3072 2.9315I 1.2712 3.2629 1.1275 3.59Q'4 0.8145 3.9063 0

SM

!(CTLi)

1 4 2 4 2 4 2 4 1

0.3431 2.6119 1.8958 4.7518 2.6145 5.0846 2.2550 3.2580 0 22.8148

;:: 0.7605

The required value is CTLi = 0.8309 The value of fJi for the next iteration is given by: 0.800 = 0.7867 0.8309 ~ 0.7605 { 5 X 0.8309

-----=--=-=-""':".~. ~."...,.=.,.,-

1+

Hydrodynamic pitch angles-second iteration (with fJi

cl

x

tanfJ

fJo

0.2 0.3

0.9534 0.7099

43.6331 35.3703

[ I-ill

1 - w(x) 1.1412 1.0799

]~

= 0.7867)

tuf

taofJI

fJ!

fJ'1 - fJo

1.09Q 1.080

1.5075 1.0524

56.4418 46.4617

12.8087 11.0914

_


266 [ I-ill 1- w(x)

x

tanjJ

jJo

0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.5717 0.4790 0.4117 0.3618 0.3215 0.2898 0.2647

29.7580 25.5931 22.3779 19.8887 17.8218 16.1596 14.8280

]!

1.0421 1.0183 1.0026 0.9903 0.9826 0.9758 0.9685

x

Ar

t;,

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.3015 0.3157 0.3241 0.3270 0.3243 0.3188 0.3116 0.3041 0.2966

1.0558 1.0082 0.9917 0.9851 0.9771 0.9543 0.9001 0.7519 0

(31 - (30

tuf

tanlh

(31

'1.070 1.055 1.030 1.000 0.970 0.940 0.910

0.8103 0.6541 0.5405 0.4554 0.3895 0.3379 0.2966

39.0189 33.1881 28.3900 24.4838 21.2814 18.6681 16.5198

~Ut

VR

VA

~

0.2677 0.2409 0.2041 0.1675 0.1308 0.0976 0.0716 0.0503 0.0328

1.4131 1.6953 1.9883 2.2946 2.6122 2.9301 3.2614 3.5896 3.9058

dGTLi d:p

9.2609 7.5950 6.0121 4.5951 3.4596 2.5085 1.6918

8M

f(GTLi)

1 4 2 4 2 4 2 4 1

0.3533 2.7232 2.0011 5.0697 2.8195 5.5634 2.5069 3.7042 0

0.3533 0.6808 ' 1.0006 1.2674 1.4098 1.3908 1.2535 0.9261 0

.,

I

24.7414

CTLi

For

7]i

For

7]~

= =

=

1

3" x 0.1 x 24.7414

0.8000,

GTLi

0.7867,

GTLi

= =

.=

0.8000

0.8247

0.7605 0.8247

By linear extrapolation, for the required CT Li 7],

=

= 0.8309

0.7867 - 0.8000 x ( 0.8309 _ 0.7605 ) 0.7605

+ 0.8247 -

Hydrodynamic pitch angles-third iteration (with

x

tan (3

(30

0.2

0.9534

43.6331

1-w [ 1- w(x) 1.1412

]!

7]i

=

0.7854

= 0.7854)

tuf

tan(3r

jJ'[

(31 - {3?

1.090

1.5100

56.4854

12.8523

'c' lJ

~

Propeller Design x

tan,8

,80

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.7099 0.5717 0.4790 0.4117 0.3618 0.3215 0.2898 0.2647

35.3703 29.7580 25.5931 22.3779 19.8887 17.8218 16.1596 14.8280

[ I-ill 1- w(x)

1.0799 1.0421 1.0183 1.0026 0.9903 0.9826 0.9758 0.9685

x

)..1

I\,

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.3020 0.3162 0.3245 0.3276 0.3248 0.3193 0.3121 0.3046 0.2971

1.0561 1.0084 0.9916 0.9850 0.9770 0.9541 0.8998 0.7515 0

r

267

tuf

tan ,81

,8'j

,8'j - ,80

1.080 1.070 1.055 1.030 1.000 0.970 0.940 0.910

1.0541 0.8116 0.6552 0.5414 0.4561 0.3901 0.3385 0.2971

46.5090 39.0652 33.2315 28.4297 24.5196 21.3135 18.6969 16.5457

11.1387 9.3072 7.6384 6.0518 4.6309 3.4917 2.5373 1.7177

~Ut

VR

VA

ltA

dCTLi dx

8M

f(CTL;)

0,3543 0.6835 1.0056 1.2749 1.4197 1.4029 1.2650 0.9383 0

1 4 2 4 2 4 2 4 1

0.3543 2.7340 2.0113 5.0995 2.8395 5.6116 2.5300 3.7530 0

I -

!

0.2688 1.4129 0.2421 1.6950 0.2053 1.9 88 4 0.1686 2.2944 0.1318 2.6120 0.0985 2.9299 0.0723 3.2613 0.0510 3.5895 0.0334 . 3.9057

24.9332

=

CTLi

1

3" x 0.1 x 24.9332

=

0.8311

This is almost exactly what is required, and no further iterations are necessary Design of blade sections and cavitation checks CL

~ = ~ XI\, sin,81 tan(,81 -,8)

c D from Table 9.5 (Morgan) t

D

i

L .,. .

',.

=

to

(1 - x) D

+x

tl D

= (1 -

x) x 0.060

+ x x 0.003 =

0.060 - 0.057 x

268


{JI

,,;

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

{3'j - {30

56.4854 12.8523 46.5090 11.1387 39.0652 9.3072 33.2315 7.6384 28.4297 6.0518 24.5196 4.6309 21.3135 3.4917 18.6969 2.5373 16.5457 1.7177

Lc =

CL

K,

1.0561 1.0084 0.9917 0.9850 0.9770 0.9541 0.8998 0.7515 0

-

0.08415 0.09050 0.08580 0.07581 0.06197 0.04702 0.03344 0.02012 0

D

0.2110 0.2336 0.2538 0.2708 0.2832 0.2883 0.2805 0.2445 0

0.0486 0.0429 0.0372 0.0315 0.0258 0.0201 0.0144 0.0087 0.0030

CL

0.2303 0.1836 0.1466 0.1163 0.0911 0.0697 0.0513 0.0356

0.3988 0.3874 0.3381 0.2799 0.2i88 0.1631 0.1192 0.0823

X

12.6028ms- 1

V cos({3/ - (3) A ',q smfJ PA

\

-ct

(a = 0.8 mean line)

0.06790CL

Y.4. = [l-w(x)]V = [l-w(x)] Vn =

t

c D

c

D

+ Pg ( h -

x R) - PV ~pVfi

=

101.325 + 1.025 X 9.81 (6.50 - 3.5 x) - 1.704 2 1 2 X 1.025 X Vn

=

164.9801 - 35.1934x 0.5125 Vfi

.

,

., .i

and pendix7. Q max

X

Qml nl

f C

0.2 0.3 0.4 0.5 0.6

0.0271 0.0263 0.0230 0.0190 0.0149

the limits of the cavitation free "bucket" are obtained from Ap

Q?

,

1- w(x)

0.6142 0.5966 0.5206 0.4311 0.3370

0.5989 0.6689 0.7183 0.7522 0.7759

\0l.

Vn

7.5478 10.6643 8.4300 14.2889 9.0526 17.9984 9.4798 21.7505 9.7785 25.5415

-Cpm1n 2.7098 1.4758 0.9089 0.6079 0.4303

Q~nax

10.2370 6.1553 3.9466 2.6202 1.8030

.,

Q~nin

-8.6027 -3.9016 -1.8831 -0.8732 -0.4736

1 ~ l I

i

!

Propeller Design x

f C

0.7 0.0111 0.8 0.0081 0.9 0.0056 1.0

,

269

0'':'

1- w(X)

0.2512 0.1836 0.1267

0.7954 0.8078 0.8191 0.8315

VR

-Cprn1h

a~ax

a~in

29.3701 33.2018 37.0542 40.9287

0.3175 0.2422 0.1894 0.1512

1.1770 0.7809 0.4997

-0.2034 -0.1328 -0.1043

VA 10.0243 10.1805 10.3230 10.4792

The values of O'i lie between a max and amin, leaving a reasonable margin for variations in angle of attack due to circumferential variations in wake at each radius. Thrust loading and power coefficients tan'Y

=

0.008 CL i

x

\

(]O

0.2 43.6331 0.3 35.3703 0.4 29.7580 0.5 25.5931 0.6 22.3779 0.7 19.8887 0.8 17.8218 0.9 16.1596 1.0 14.8280

dCTL dx

=

~

dCp dx

=

dCTLi tan(Jj + tan'Y dx tan(J

dCTLi

(1- tanfh tan'Y)

(JI

tan'Y

'Yo

dCTLi dx

dCTL dx

dCp dx

56.4854 46.5090 39.0652 33.2315 28.4297 24.5196 21.3135 18.6969 16.5457

0.02006 0.02063 0.02367 0.02858 0.03656 0.04905 0.06711 0.09721

1.1492 1.1818 1.3556 1.6369 2.0938 2.8081 3.8396 5.5521

0.3543 0.6835 1.0056 1.2749 1.4197 1.4029 1.2650 0.9383 0

0.3436 0.6686 0.9863 1.2510 1.3916 1.3715 1.2319 0.9074 0

0.5686 1.0348 1.4693 1.8200 1.9928 1.9591 1.7993 1.4106 0

8M f(CTL) f(Cp) 1 4 2 4 2 4 2 4 1

0.3436 0.5686 2.6745 4.1392 1.9726 2.9385 5.0041 7.2798 2.7832 3.9857 5.4860 7.8362 2.4638 3.5986 3.6297 5.6426 0 0

Totals 24.3575 35.9891

= '13 x 0.1 4 .

I I

~

Cp =

1

"3

x 24.3575 = 0.8119

x 0.1 x 35.9891 = 1.1996

The value of CTL finally obtained is different from the initial value of 0.8178 by less than one percent. Hence, no iteration for CTL is necessary.


270 The delivered power: 1

-3

= Cp2PAoVA

= 1.1996 x 0.5 x 1.025 x 38.4845 x 9.83023

= 22476kW The propeller efficiency in the behind condition: 11B =

=

0.8119 = 0.6768 1.1996

Lifting surface corrections

Z

AE

=6

Ao

to = 0.060 D

= 0.775

<~

Corrected

£e = kc x £e

Corrected a~

"

= kcx X a:f to

a~ = k t D x 57.3

= k t x 0.060 x 57.3 = 3.438 k t

kef kcx and k t from Appendix 8.

P D =

1rX

tan
Corrected

Uncorreeted

x

/31

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

56.4854 46.5090 39.0652 33.2315 28.4297 24.5196 21.3135 18.6969 16.5457

fie 0.3020 0.3162 0.3245 0.3276 0.3248 0.3193 0.3121 0.3046 0.2971

0.6142 0.5966 0.5206 0.4311 0.3370 0.2512 0.1836 0.1267

0.0271 0.0263 0.0230 0.0190 0.0149 0.0111 0.0081 0.0056

2.0390 2.6057 2.7543 2.4742 2.2124 1.8584 1.3108 0.2995

2.2262 1.5432 1.1944 1.1500 1.1759 1.2720 1.4475 1.9178

1.2936 0.9155 0.6492 0.4499 0.2982 0.1871 0.1167 0.0823

1.2524 1.5546 1.4339 1.0666 0.7456 0.4668 0.2407 0.0379

4.4474 3.1475 2.2319 1.5468 1.0252 0.6432 0.4012 0.2829

Propeller Design

271 Corrected

x

PID

fie

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

62.1852 51.2111 42.7310 35.8449 30.2005 25.6296 21.9554 19.0177 16.5457

1.1910 1.1727 1.1609 1.1348 1.0971 1.0550 1.0132 0.9745 0.9333

0.0603 0.0406 0.0275 0.0218 0.0175 0.0141 0.0117 0.0107

x

P D

e

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1.1910 1.1727 1.1609 1.1348 1.0971 1.0550 1.0132 0.9745 0.9333

Final blade geometry

f

D

t D,'

0.2110 0.2336 0.2538 0.2708 0.2832 0.2883 0.2805 0.2445 0

0.04'86 0.0429 0.0372 0.0315 0.0258 0.0201 0.0144 0.0087 0.0030

0.0603 0.0406 0.0275 0.0218 0.0175 0.0141 0.0117 0.0107

e

(It is necessary to "fair" the values of PID and The blade sections are to have the NACA a (modified) thickness distribution.

=

fie)

0.8 mean line and the NACA 66

As Example 7 illustrates, the design of a propeller using the circulation the ory involyes considerable computation. It is therefore customary nowadays to use computers for propeller design based on circulation theory. When computers are used, it is possible to adopt further refinements in the de sign procedure such as the use of the Lerbs induction factors instead of the Goldstein factors. It is also possible to modify the initial design obtained, 4

I

~-"


272

by using the lifting line theory to take the lifting surface effects into ac count more accurately than is possible using the correction factors described here. Modern computer aided propeller design procedures use Computa tional Fluid Dynamics techniques such. as surface panel methods. As mentioned earlier, there are several different approaches to the design of propellers using the circulation theory. Anyone of these approaches may be adopted, including the various corrections that it involves, but it is necessary to verify these propeller designs by model experiments in a cavitation tunnel at least until confidence in the design procedure has been developed.

Problems 1. A twin screw ship with a geared steam turbine propulsion plant has a de sign speed of 35 knots at which the effective power is 16000 kW. The wake fraction, thrust deduction fraction and relative rotative efficiency based on thrust identity are 0.000, 0.050 and 0.990 respectively. The shafting efficiency is 0.940 and the estimated blade area ratio required is 0.800. The propellers are to have three blades and a diameter of 3.5 m. Determine the total shaft power of the propulsion plant, the propeller rpm and the pitch ratio of the two propellers using the Gawn Series data given below. Do you think that the results are correct? Explain your answer.

[~~r \

0.3000' 0.3500 ' 0.4000 0.4500 0.5000

AD = 0800 Ao . .

AD = 0.650 Ao

= 0.950

AD Ao

J

PjD

170

J

PjD

170

J

PjD

170

1.4364 1.2313 1.0892 0.9817 0.8961

1.7026 1.4875 1.3490 1.2485 1.1707

0.8390 0.8092 0.7801 0.7520 0.7253

1.4560 1.2433 1.0998 0.9924 0.9061

1.7290 1.5024 1.3620 1.2625 1.1832

0.8354 0.8044 0.7750 0.7469 0.7201

1.4858 1.2556 1.1090 1.0025 0.9157

1.7747 1.5176 1.3721 1.2748 1.1946

0.8325 0.8000 0.7704 0.7422 0.7155

2. A twin-screw ship has two steam turbines each producing a shaft power of

11500 kW at 3000 rpm. The turbines are connected to the two three-bladed propellers through reduction gearing of ratio 12:1. The shafting efficiency is 0.940. The effective power of the ship is as follows: Speed, knots Effective Power, kW :

30.0 9329

32.0 11693

34.0 14457

36.0 17658

38.0 21337

40.0

25533

Propeller Design

273

The wake fraction, thrust deduction fraction and relative rotative efficiency based on thrust identity are 0.000, 0.050 and 0.990 respectively. The depth of immersion of the propeller axes is 3.2 m. Design the propellers using the Gawn Series data given in Problem 1 and the Burrill cavitation criterion for warship propellers.

!"

h i I, f"

i 1 I

3. A coaster is to have an engine of 600 kW -brake power at 900 rpm connected to the propeller through 5:1 reduction gearing. The shafting efficiency is 0.950. The effective power of the vessel is as follows: Speed, knots Effective power, kW:

6.0

8.0

10.0

12.0

14.0

26.8

75.5

168.6

325.0

566.1

The wake fraction, thrust deduction fraction a!nd relative rotative efficiency based on torque identity are respectively 0.270,0.250 and 1.010. The propeller shaft axis is 1.8m below the waterline. Design the propeller using the data of Table 9.1 and determine the maximum speed of the vessel. 4. A twin screw ship with geared steam turl,:>ine machinery has three-bladed propellers of diameter 3.5 m, pitch ratio 1.5 and blade area ratio 0.85. The open water characteristics of the propellers are as follows: J

1.2000

1.2500

1.3000

1.3500

1.4,000

KT

0.2040

0.1780

0.1523

0.1270

0.1022

0.4956

0.4399

0,3854

0,3322

0.2803

10KQ

:

At various speeds V, the effective power PE, the wake fraction w, the thrust deduction fraction t and the relative rotative efficiency 1JR (based on thrust identity) are as follows: V, knots:

15.0

20.0

25.0

30.0

35.0

40.0

PE, kW:

825

2257

4928

9329

16000

25533

w

0.040

0,030

0.020

0.010

-0.010

0.010

t

0.070

0.065

0.060

0.055

0.050

0.055

TJR

0.995

0.995

0.994

0.992

0.990

0.992

The shafting efficiency is 0.940. Determine the shaft power and rpm of the turbines as a function of ship speed, the gear ratio between the turbines and

I

L_

274

Basic Ship Propulsion the propellers being 12:1. Determine also the ship speed at which the turbines run at their rated speed of 3000 rpm and 'the corresponding shaft power.

5. A single-screw ship has an engine of brake power 5000 kW at 126 rpm directly

connected to the propeller of diameter 5.200 m, pitch.ratio 0.8430 and blade

area ratio 0.502. The shafting efficiency is 0.970. The effective power of the

ship in the clean naked hull condition is as follows:

Speed, knots Effective power, kW:

10.0

12.0

14.0

16.0

544.5

1031.2

1769.2

2823.3

The open water chara:eteristics of the propeller may be obtained from Table 9.2. After the ship has been in service for a year, its effective power is found to have increased by 30 percent over the clean naked hull condition, and the wake fraCtion, thrust deduction fraction and relative rotative effi ciency based on thrust identity are 0.250, 0.200 and 1.020 respectively. The roughening of the propeller blade surfaces causes a decrease in KT of 0.0012 ana an increase in K Q of 0.00013. Determine the maximum speed of the ship in this condition and the corresponding brake power and engine IiPm if the maximum torque of the engine is not to be exceeded. . 6. A single screw tug has an engine of 900 kW brake power at 600 rpm connected

to the propeller through a 4:1 reduction gearbox. The shafting efficiency is

0.950. The wake fraction is 0.200, while the thrust deduction fraction varies linearly with speed with values of 0.050 and 0.180 at zero speed and 12 knots respectively. The relative rotative efficiency is 1.000. The effective power PE of the tug at different speeds V is as follows:

, V, knots:

8.0

9.0

10.0

11.0

12.0

13.0

14.0

15.0

PE, kW:

68.9

104.1

150.6

210.2

285.0

377.1

488.8

622.3

Design the propeller for maXimum bollard pull. Calculate the free running speed of the tug and the corresponding brake power and rpm of the engine. Compare the results obtained with this design and the results of Example4. Use the data of Table 4.3. 7. Design the propeller for the tug in Problem 6 for maximum free running speed,

and determine the bollard pull and the corresponding brake power and engine

rpIll. Compare these results with those of Problem 6. The optimum propeller

diameter is 3.0 m.

'/i

Propeller Design

.275

8. A single screw fishing trawler has an engine of 1150 kW brake power at 240 rpm directly connected to the propeller. The shafting efficiency is 0.970. The effective power of the trawler is as follows: Speed, knots Effective poWer, kW:

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0.44

5.99

27.24

82.30

191.31

381.11

682.51

The wake fraction is 0.180 and the relative rotative efficiency 1.030 based on thrust identity. The thrust deduction fraction varies linearly with speed being 0.075 at 4.0 knots and 0.150 a.t 14.0 knots. The depth of immersion of the propeller shaft axis is 2.5 m. Design the propeller for full power absorption at the design trawling speed of 4.0 knots, and determine the maximum pull that can be exerted on the trawl gear at this speed. Determine also the free running speed of the trawler and the corresponding brake power. If the resistance of the trawl gear when full of fish is as given in the following table, calculate the maximum speed of the trawler with a full catch and the corresponding brake power and engine rpm. Speed, knots Trawl resistance, kN:

2.0

4.0

6.0

8.0

10.0

18.66

69.64

150.47

259.92

397.16

Use the data of Table 4.3 and check that the expanded blade area ratio of 0.500 is adequate with the help of Eqn. 9.2. 9. A twin-screw tug has two engines each of brake power 500kW at 1200rpm connected to the propellers through a 4:1 reduction gearbox. The propellers are of diameter 2.0 m, pitch ratio 0.700 and blade area ratio 0.500, their Op<'lJ water characteristics being as given in Table4.3. The effective power of i h tug at various speeds is as follows: Speed, knots Effective power, kW:

2.0

4.0

6.0

8.0

10.0

12.0

14.0

0.04

5.32

23.57

67.74

153.65

300.00

528.23

The wake fraction is 0.100 and the relative rotative efficiency is 0.980 based on torque identity. The thrust deduction fraction varies linearly with speed, being 0.040 at zero speed and 0.100 at 12.0 knots. Determine the maximum towrope pull and the corresponding brake power and engine rpm as a function of speed. Also calculate the speed at which the engines develop the maximum brake power and the free running speed of the tug. The maximum rated torque and rpm of the engines are not to be exceeded.

L _

276


10. Design the propellers for a twin-screw liner using the lifting line ~heory with lifting surface corrections. The ship has a design speed of 35.0 knots at which the effective power is 16000 kW. The three-bladed propellers are to have a diameter of 3.5 m and to run at 300 rpm. The propeller shaft centre lines are 3.2 m below the waterline. The wake fraction is 0.000, the thrust deduction fraction 0.050 and the relative rotative efficiency 0.990 based on thrust iden tity. The wake may be assumed to be uniform. The expanded blade area ratio is estimated to be 0.850 and the blade outline is to be according to Morgan, Silovic and Denny, Table 9.4, with no skew. The blade thickness fraction is to be 0.050 and the tip thickness 0.003 times the propeller diameter with a linear thickness variation. The blade sections are to have NACA a = 0.8 mean lines and NACA-16 (modified) thickness distribution. Calculate the required shaft power of the propulsion turbines if the shafting efficiency is 0.940.

~

'.•...' •.

.~

'

:,"

(J:.1

1

j.

. "i;'1\ .{' ., . "

!.,

I'

" CHAPTER

10

Ship Trials and Service Performance 10.1

/

Introduction

As the construction of a ship nears completion , and just after it is complete, a wide range of tests and trials are carried out on the ship. These tests and trials are carried out for quality assurance:purposes, to check that the vari ous systems in the ship function properly and to see that the ship meets the requirements of the contract between the ship owner and the shipbuilder. Tests and trials may be divided into ~ shop tests for testing system com ponents, (b) installation tests to see that the systems installed in the ship function as required and meet their specifications, (c) dock trials to observe the operation of the various ship systems in typical operating conditions, and finally (d) sea trials to observe the performance of a ship at sea. Dock trials, which are carried out after the ship has been launched and is nearly complete and after the installation tests are over, establish that the propulsion plant and its auxiliaries are ready for sea trials~ea trials are often carried out twice: (i) the builder's trials to observe the performance of the ship at sea, to identify any deficiencies and determine how they are to be overcome, and (ii) acceptance trials to demonstrate that the ship meets the contractual require ments. Sea trials include a number of tests that can only be carried out at sea: speed trials, economy power tests, manoeuvring trials, anchor windlass tests, distillation plant ,tests and calibration of navigation equipment. Dock ,./

Basic S4ip Propulsion

278

trials and speed trials are basically tests of the ship propulsion system. For certain types of ships such as tugs, it is necessary to determine the maximum static pull that the ship can exertj bollard pull trials are carried out for this purpose. The service performance of a ship is routinely monitored to ensure that it continues to perform efficiently and to decide what remedial actions are to be taken and when. The analysis of service performance data on speed and power can result in several benefits: - The effect on speed and power of the worsening hull and propeller surfaces can be determined and used to decide upon the most advan tageous drydocking, cleaning and painting schedules as well as the time when the propeller must be replaced by one of lower pitch. ' - The effect of weather (wind and waves) on ship speed and power can ,be determined, and this information may be used to decide the optimum route for a ship in given weather conditions. . . I

The analysis of ship trials and service performance data is also useful iti the design of future ships. ;

10.2

Dock Trials

In dock'trials, the ship is secured to the pier by mooring lines and the propulsion plant run. Since the ship is stationary, the propeller runs at 100 percent slip, so that the thrust and torque coefficients are much higher than they woule be in normal operating conditions. The maximum speed (rpm) at which the engine may be allowed to run is limited by the maximum torque or thrust that is permissible, the allowable loads in the mooring system and the disturbance created by the propeller in the dock or basin in which the dock trials are carried out. Example 1 A ship has an engine of rating 5000kW brake power at 120 rpm directly connected to a propeller of 5.0 m diameter and 1.0 pitch ratio. Determine the maximum rpm at which the engine may be run in the dock trials if the maximum rated torque of

Ship Trials and Service Performance

279

the engine is not to be exceeded. The propeller has a KQ of0.0600 at J = O. The shafting efficiency is 0.970 and the relative rotative efficiency 1.030 (thrust identity). i

i'

PE

=

5000kW

D

=

5.000m

P D

=

1.000

TJs

=

0.970

TJR

=

1.030

PD

=

PE TJs

n = 120 rpm

=

5000

X

=

2.0s- 1

KQ

0.970

=

= 0.0600

atJ

=0

4850 kW

Propeller torque at engine rated power: Q

=

PD -271"n

2

=

Qr/R pD5KQ

n

n

10.3

271"

1.43288- 1

4850 X 2.000

= =

=

385.951 kN m

385.951 x 1.030 1.025 x 5.0 5 X 0.0600

=

2.0684

86.29 rpm

Speed Trials

The speed trials of a ship may have many objectives: - to determine the relationship between ship speed, engine power and propeller rpm at a specified displacement and trim with the ship hull clean and newly painted and the propeller surfaces clean; - to see that the ship meets its contract requirements of speed, power, rpm and fuel consumption; - to determine the correlation between the speed-power-rpm relationship predicted from model tests and that obtained in the speed trials.

280


The speed trials may also be used to calibrate speed logs (instruments to determine the speed of the ship by measuring the relative velocity of water close to the hull) and to determine the relation between ship speed and propeller rpm for use-in navigation. Both the speed log calibration and the " speed-rpm relationship depend upon the condition of the hull surface and therefore change with time. The basic procedure for carrying out speed trials is to run the ship sev eral times in opposite directions over a known distance (usually one nautical / mile) and measure the time taken to traverse this distance along with mea surements of propeller rpm, power and, if possible, thrust. Runs in opposite directions are necessary to eliminate the effect of currents in the water and obtain the true speed of the ship through the water. It is important to carry out the speed trials in good weather with low winds and a calm sea. The "me~ured mile" over which the speed trials are carried out must fulfil certain requirements: - The depth of water on the trials course should be sufficient to ensure that there are no shallow water effects. A depth of water exceeding twenty times the draught of the ship is normally recommended. For . high speed ships even this may be insufficient and a criterion involving ship speed must be used, e.g. It 2: 30Fn T, where h is the depth of water, Fn the Froude number and T the draught. - The measured mile should be close enough to the shore to allow the range beacons that define the measured mile to be clearly observed in good weather. - There should be sufficient space for the ship to turn at the end of a run in either direction and attain a steady speed during the approach to the measured mile in the opposite direction. The length of approach required depends upon the size and speed of the snip and must be sufficient to allow 99.8 percent of the required speed to be attained before the ship reaches the start of the measured mile. - The trials area should be sheltered, and be close to a good anchorage and near shipbuilding centres.

281

Sbip Trials and Service Performance

- It should be possible to ensure that there is no other traffic in the area during the trials. - The current in the water should be small and not subject to sudden and large changes in magnitude and direction. A measured mile meeting all these requirements is difficult .to find, particu larly for very large ships. Since the measured mile is required basically for determining the speed of the ship accurately, al~ernative methods of speed measurement are necessary if one is to dispense with the measured mile. Two methods which are increasingly being adopted today are the use of shore based radio positioning systems and satellite navigation systems which allow the position of a ship at any instant to fbe deter~ned with sufficient accuracy: The use of such systems allows speed trials to be carried out in water of adequate depth, with the direction ,of the runs chosen to minimise the effect of the wind and waves, and at lo~ations close to the shipyard.

.

,

i

l

Speed trials may be carliied out only at ~he highest speed to demonstrate that the ship meets its contractual obligations. However, it is more useful to carry out "progressive speed trials" in which groups of runs are made at' engine power settings between half full power and full power. At least four engine settings should be used, 'more if the speed of the ship is very high. There should be a minimum of two runs, one in each direction at each engine setting, preferably-three and even four or five at the highest power. However, the trials should be completed without interruption, and there should be no break in the sequence of runs at anyone engine setting so that the runs in each group occur at approximately equal intervals of time. Just before the speed trials,'the ship is surveyed to see that the propeller is clean and polished and the hull"is in the standard clean, newly painted condition. The hull roughness is measured. The weather for the trials should be good with a minimum of wind and waves in the sea. When the ship is ready for the trials to commence and the engine has warmed up, the ship proceeds on the prescribed course. As the approach buoy is reached (2-3 miles from the start of the measured mile), the ship is steadied on the course, the engine rpm is fixed and the rudder is used as little as possible so that a constant speed is attained by the beginning of the measured mile. At the end of the measured mile, the ship turns slightly

..•§.~'-----

282


REAR BEACON

FRONT BEACON

+I

REAR BEACON

I I I

"J

t

FRONT BEACON

• i.'

,I.

I

I

I I,

APPROACH B~OY

I I

APPROACH

B~OY

~I"'l~

~ .MEASURED ~

MILE

Figure 10.1: Measured Mile.

and proceeds some two or three miles further before turning to traverse the measured mile in the opposite direction, as shown in Fig. 10.1. The following are the observations made.in the speed trials:

•

- the ,draughts of the ship forward and aft just before and just after the

trials;

- the temperature and density of the water on the trials course; - the condition of the seaj - in each run: - the direction of the run; - the time of day at the start of the measured mile; - elapsed time over the measured mile, or .the speed of the ship;'

'/P


283

- the average propeller rpm, or the total number of revolutions over the measured mile; the corresponding engine power; the relative wind velocity and direction; the propeller thrust, if possible; - large rudder angles, if any; - the consumption of consumables and the addition of water ballast and its redistribution. I

If the trials are carried out on a measured mile, the time over the mile is measured independently by three or four opservers who start their stop watches when the rear and forward beacons at the start come in line and stop them when the beacons at the end of tl1e measured mile come in line. The readings of the independent obaervers should agree. Radio positioning and satellite navigation systems allow the time taken to traverse a given distance to be automatically recorded. The engine power may be measured by il1dicators fitted to each cylinder of the main (diesel) engine. In,dicators give the indicated power, and to obtain the brake power the mechanical efficiency determined in the engine test-bed trials must be used. It is preferable to measure the power by a torsionmeter fitted on the propeller shaft. It is necessary to correct the torsionmeter readings for the residual torque in the propeller shaft. The "torsionmeter zero" is determined in turbine ships by allowing the shaft to coast to a stop after running at a moderate speed ahead and then astern and taking the average of the minimum ahead torque and the minimum astern torque as the zero reading. In diesel ships, the propeller shaft is turned first ahead and then astern by the engine turning gear, and the zero reading taken as the average of the readings after the ahead and the astern revolutions. In a ship fitted with an electrical drive, the power may be determined from the power consumption of the drive motor. Finally, the measured power must be multiplied by an estimated shafting efficiency to obtain the delivered power. Ships are not usually fitted with thrust meters, but thrust measurements are necessary for a complete analysis of the trial results. Appropriate in struments such as a revolution counter, a wind vane and an anemometer are provided to measure the other parameters..


284

Running plots of Ps/n 3 versus V/n and of V/n versus the time of day at mid-run are made during the trials to check for errors in measurement or in recording the data. Ps is the shaft power, n the propeller revolution rate and V the ship, speed. The ship speed obtained from the measurement of the elapsed time over the measured mile is the speed over the ground. What is needed, however, is the speed of the ship through the water, i.e. the speed over the ground corrected for the speed of the current in the water. When three or more runs in opposite directions are made at a constant power setting, the correct speed through the water is usually obtained by determining the "mean of means" of the individual values of speed, as illustrated in the following example. Example 2 Four runs in opposite directions are made by a ship over a measured mile at constant power and the following times are recorded. Determine the speed of the ship through the water at this power and the speed of the current to the North in each run.

1

Run No. Direction Time over mile, sees

Run No.

1

Direction

N

2

N S 208.0 . 158.8

Time

Va

s

knots

208.0

17.308

3 N 203.1

Mean of Means

1

2

4 S 167.2

Speed through water, Vw knots

Speed of Current, Va knots -2.695 N

19.9890

2

S

158.8

20.003

20.1975 3

N

203.1

-2.667 N

20.0932

22.670

19.9128

17.725

-2.278 N

19.6280

4

S

167.2

. -1.528 N

21.531

Speed over ground, Va =

. 3600 .. knots TIme 10 sees

Speed of current to the North, Va = ±( Va - Vw ) +for N runs, -for S runs.

285


The use of the limean of means" method to determine the speed through the water is based on the assumption that the speed of the current Vc can be expressed as a polynomial:

(10.1) where i 'is the time measured from some fixed instant, the degree of the polynomial being one less than the number of speed readings. Thus, if there are three values of the speed over the ground, it is assumed that Vc = ao + ali + a 2 f2. It can be shown that if the intervals between the runs are equal the mean of means gives the correct speed through the water. This is the reason for the requirement that all the runs for a particular engine setting be completed in an unbroken sequence wit40ut interruption. In tria~ in which only two runs, one in each direction, are made at each engine setting, and in cases where the time interval between consecutive runs is far from being constant, a different metho,d of correcting for the effect of the current in the water may be used. In thi~' method two curves of V In, one for each direction, are drawn separately as;a function of the time of day at mid-run, and a mean between the two curves is taken to give the corrected speed through the water. The procedure is used in the next example, which illustrates a method of analysing speed trials data. Example 3 The following are the observations made in the speed trials of a tanker in which two runs in opposite directions were taken at each engine setting:

(':'

Run No. Direction Time at start, hr-min: Time on mile, sees Total revolutions Shaft power, kW

1 E 07-46 277.2 397 4556

2 W 09-19 252.6 363 4592

3 E 10-01 247.2 413 7908

4 W 10-48 223.8 369 7996

Run No. Direction Time at start, hr-min: Time on mile, sees Total revolutions Shaft power, kW

7 E 13-28 213.6 399 11484

8 W 14-03 212.4 398 11498

9 W 16-02 210.0 421 13952

10 E 16-45 198.0 396 14154

5 E 11-47 223.8 407 10626

11 W 17-56 201.0 422 16890

6 W 12-22 213.0 380 10552

12 E 18-32 197.4 412 16742

286


The ship has a propeller of 6.5 m diameter and 0.8 pitch ratio, whose open water characteristics are as given il,l Table 4.3. The effective power of the ship estimated from model tests is as follows: Speed, knots Effective power, kW:

12.0 1530

14.0 3062

16.0 5584

18.0 9486

20.0

15241

Analyse these trial trip data.

~.

The calculations are carried out in the following table: Run No. 1 Direction E Start time, hr-min: 07-46 Time on mile, s: 277.2 Total revolutions 397 Shaft power, Ps kW: 4556 Speed over ground, knots: 12.987, ms- 1 : 6.6805 Va' s-l: 1.4322 Revolution rate, n rpm: 85.93 m: 4.6645 VaIn hr: 7.8052 Time at mid-run, Mean Vln m: 4.9147 Speed through water, ms- 1 : 7.0388 knots: 13.684 Vw Speed of current, ms- 1 : -0.3583 Ve (East), knots: -0.697

2

3

W

E

09-19 252.6 363 4592 14.252 7.3311 1.4371 86.22 5.1013 9.3518 4.7967 6.8933 13.401 -0.4378 -0.851

10-01 247.2 413 7908 14.563 7.4913 1.6707 100.24 4.4839 10.0510 4.7756 7.9786 15.510 -0.4873 -0.947

4 W 10-48 223.8 369 7996 16.086 8.2745 1.6488 98.93 5.0185 10.8311 4.7607 7.8494 15.259 -0.4251 -0.826

5 E 11-47 223.8 407 10626 16.086 8.2745 1.8186 109.12 4.5499 11.8144 4.7425 8.6247 16.767 -0.3502 -0.681

6

W 12-22 213.0 380 10552 16.901 8.6914 i 1.7840 107.04 4.8734 12.3962 4.7279 8.4346 16.397 -0.2595 -0.504 I

'!

I '-:-'

'If

'~

I~~~4 >'iW~

I

~,

~'~".~

, :t';\

~{-~

f~

~rr

j"1

~iJ :\"7}'

f",

'"l

~.~; """ <{}l ;

Run No. 7 8 Direction E W Start time, 14-03 hr-min: 13-28 Time on mile, s: 213.6 212.4 Total revolutions 399 398 Shaft power, Ps kW: 11484 11498 Speed over ground, knots: 16.854 16.949 ms- 1 : 8.6697 8.7186 Va S-I: 1.8680 1.8738 Revolution rate, n rpm: 112.08 112.43 m: 4.6412 4.6529 VaIn Time at mid-run, hI: 13.4963 14.0795

9 W 16-02 210.0 421 13952 17.143 8.8183 2.0048 120.29 4.3986 16.0625

10 E 16-45 198.0 396 14154 18.182 9.3527 2.0000 120.00 4.6764 16.7775

11 W 17-56 201.0 422 16890 17.910 9.2131 2.0995 125.97 4.3882 17.9612

12 E 18-32 197.4

412 16742 18.237 9.3812 2.0871 125.23

4.4948

18.5608

~

j;,. r'H!

~i~

:~'~:~ '~

::~~

'~l~

:~:t~

>L ;:ip

l~

;~i ,.~

'il\1 .~ .~

iii:.

Ship Trials and Service Performance 1fean Vln Speed through water, Viv Speed of current, Va (East),

m: ms- 1 : knots: ms- 1 : knots:

4.6878 8.7568 17.023 -0.0871 -0.169

287 4.6599 8.7317 16.975 0.0131 0.025

4.5482 9.1182 17.726 0.2999 0.583

4.5133 9.0266 17.548 0.3261 0.634

4.4848 9.4158 18.305 0.2027 0.394

4.4930 9.3773 18.230 0.0039 0.008

In this table, the speed over the ground Va is obtained by dividing the distance 1 nautical mile by the time over the measured mile, the propeller revolution rate n by dividing the total revolutions over the measured mile by the time over the measured mile, and the time at mid-run by adding half the time on the measured mile to the time of day at the start of the meas\lred mile, taking care of the appropriate units in each case. The values of Vain are plotted as a/function of'the time at mid-run separately for the East runs and the West runs, and the mean curve between these two obtained. This mean curve is taken to give the value of V In corrected for the influence of the current. The speed tluough water Vw for each run is then obtained from the' mean V In curve, and the speed of the;current is given by:

Va = ±( Va

-;Vw )

the positive sign being for runs in one directibn and the negative sign for the runs in the opposite direction. The subsequent ~nalysis is carried out as given in the following table. Run No. Mean V, ms- 1 : knots: s-l: Mean n, rpm: Mean Ps , kW: kW: PD, kW: PE, 1)D (qpc) 10KQB J KT 1)0

VA wQ 1)H 1)R'

·" : "£

I: ; t:

I

'

.

ms- 1 :

1&2 3&4 6.9661 7.9140 13.542 15.385 1.4346 1.6598 86.08 99.58 4574 7952 4437 7713 4681 2636 0.5941 0.6069 0.2011 0.2257 0.5699 0.5166 0.1402" 0.1636 0.6323 0.5960 5.3143 . 5.5734 0.2371 0.2957 0.9396 1.0183

5&6 8.5296 16.582 1.8013 108.08 10589 10271 6558 0.6385 0.2352 0.4952 0.1727 0.5787 5.7980 0.3202 1.1033

7&8 8.7442 16.999 1.8709 112.25 11491 11146 7333 0.6579 0.2278 0.5119 0.1656 0.5923 6.2251 0.2881 1.1464

9 & 10 9.0724 17.637 2.0024 120.14 14053 13631 8655 0.6349 0.2272 0.5132 0.1651 0.5935 6.6796 0.2637. 1.0698

11 & 12 9.3966 18.267 2.0933 125.60 16816 16312 10136 0.6214 0.2380 0.4889 0.1754 0.5734 6.6522 0.2921 1.0837

The mean values of V, nand P s for each pair of runs have been obtained from the previous table, and:


288 PD

KQB

= P s 115 =

110 =

PD 27l"pn 3 D5

KT J KQ 2ii

PE from the PE - V curve

PE

11D =

PD

J and K T from torque indentity

VA = In.D

1}H 7]~

11D

=

7]0

When the thrust is not measured during the trials, the wake fraction may be determined only by torque identity. Also, it is not possible to determine individually the hull efficiency and the relative rotative efficiency. Howeve,r, if one adopts an assumed value of the relative rotative efficiency (or the thrust deduction fraction), for example on the basis of model test data, ~he trial results can be analysed using thrust identity also, and the individual ! components of the propulsive efficiency determined. In the method used in Example 3 for determining the correct speed through the water by obtaining a mean curve between the curves of V/ n for the two opposite directions as a function of the time-of-day, it is assumed that the ef fects of the wind on the performance of theship for the two direction~ cancel each other. The power-rpm-speed relation that is thus obtained refers to the performance of the ship in still air. However, the assumption that the effect of the wind on performance can be eliminated simultaneously with the effect of the current in the water by taking a mean between two runs in opposite directions is not strictly correct. The current in the water only affects the speed 'of the ship over the ground whereas the wind affects the resistance of the ship and hence the speed, power and propeller rpm. Moreover, partic ularly for ship-model correlation purposes, it is necessary to determine the performance of the ship for the "no air" condition which corresponds closely to the conditions of the model test. Therefore, a different and slightly more complex method may be used to correct for the effect of the wind. .The method that is generally used is to estimate the air and wind resistance RA A from the measured relative wind velocity and direction, calculate the increase in effective power D..PE, and to make a corresponding increase D..V in the speed of the ship over the ground. The air and wind resistance may be determined from model tests in a wind tunnel, or estimated by empirical methods based on such tests, e.g.: RA A

=

CD kw ~PaAT V;w

(10.2)

\


289

where:

~

CD =

drag coefficient;

kw =

'\vind direction coefficient;

Pa

=

density of air;

AT

=

equivalent transverse projected area of the ship above water equal

to 0.3 times the transverse projected area of the main hull above water plus the transverse projected area of the superstructure above the main hull;

Vrw =

relative wind velocity. I

r

The density of air is 1.226 kg per m 3 (at 15° C), The values of CD and kw depend upon the shape of the ship above water, ,typical values being: CD = 1.2 2' 3 + al () + a2 {)J-ja3 ()

kw

-

aD

= 2.769

X

10- 2

(10.3)

where: _ 0'1

= 1.000

al

a2

= -6.248

X

10-4

a3

==

2.309

X

10- 6

- () = angle in degrees of the relative wind off the bow of the ship. The method of analysing speed trials data to obtain the "no air" condition is illustrated in the following example. Example 4 ; ...

,,

! -

,-

I

The following data are obtained from the speed trials of a ship for four runs at the highest power setting: Run No. Direction Time at start .of the measured mile, hr-min: Time on the measured mile, s Average propeller rpm Average shaft power, kW Average thrust, kN Relative wind velocity, ms- 1 Relative wind direction off the bow, deg

16 N 15-49 146.8 113.7 21385 151~

17.128 10.63

17 S 16-21 155.3 112.8 21272 1495 8.281 34.05

18 N 16-55 147.5 113.3 21304 1507 18.025 16.99

19

S 17-30 152.6 112.5 21268 1492 9.576 31.95

290

Basic Ship PropulBion

The ship has a propeller of 7.0 m diameter and 1.0 pitch ratio whose open wa ter characteristics may be taken from Table 4.3. The transverse projected area of the ship above water consists of 120 m 2 of the main hull and 176.5 m 2 of the su perstructure. The effective power of the ship in the condition of the trials is as follows: Speed, knots E~ective power, k\V ;

20 6788

22 10325

24 15140

26 21532

Analyse these data. Run No.

Direction

Time at Start hr-min

Time on Mile s

15-49 16-21 16-55 17-30

146.8 155.3 147.5 152.6

16 17 18 19

N S N' S Mean of means

Run No.

16 17 18 19

Run No.

16 17 18 19

Relative Wind Speed Vrw

Observed Speed, Vo ms-~ knots 24.523 23.181 24.407 23.591

12.6147 11.9243 12.5548 12.1353

23.860

12r 2734

Wind Direction Coefficient, k w

ms-~

Relative Wind Direction, 8 deg

17.128 8.281 18.025 9.576

10.63 34.05 16.99 31.95

1.2265 1.3096 1.3014 1.3222

Increase in Effective Power due to Wind f),.PE kW

709.51 167.39 829,80 229.99 Mean of means

Slope of Effective Power Curve f),.PE/f),.\! kW/knot

2721

Wind Resistance RAA

kN

Wind Correction to Speed

56.245 14.038 66.094 18.952

Corrected Speed over Ground

f),.V knots

knots

ms-~

0.261 0.062 0.305 0.085

24.784 23.243 24.712 23,676

12.7489 11.9562 12.7119 12.1789

24.041

12.3665

Va

Ship Trials and Service Performance , j.

i';

Run No.

Propeller Revolutions

Average

n V

m- 1

n rpm

S-1

16 17 18 19

113.7 112.8 113.3 112.5

1.8950 1.8800 1.8883 1.8750

Mean of means

113.06

1.8844

291

0.152377

Run No.

Speed through Water

Speed of Current

Vw

Vo

ms- 1

knots

12.4326 12.3378 12.3925 12.3050

24.176 23.985 24.091 23.921

12.3665

24.041

Shaft Power

0.3127 0.3816 0.3194 0.1261

hr N N N N

15.837 16.372 16.937 17.521

Propeller Thrust

kW

T kN

21385 21272 21304 21268 21298

1512

1495

1507

1492

1501

Ps 16 17 18 19 Mean of means

ms- 1

Time at Mid-run

!

.

3600

\

Vo = T'Ime on ffi1'1e, S knots

kw from Eqn. (10.3) Equivalent transverse projected area above water AT

=

0.3 x 120 + 176.5

=

212.5 m2

1 A y2 k 1 1.226 5 y2 kN R AA = C D'w k 2 Pa T rw = 1.2 w x 2 x 1000 x 212. x rw

t:J..PE t:J.. V from Effective Power-Speed curve at the mean speed t:J..V =

Va

t:J..PE t:J..PE/t:J..V

Vo

+ t:J..V

292

Basic Ship Propulsion n

1.8844 12.3665

Average V =

.J

n

"IV

-

O.152377m- 1

:=

n

Average V

Va == Va - Vw for N runs

=

Vlv - Va for S runs

Time at mid-run == Time at start

+~

x Time on mile

Average values for this group of runs:

V

=

n

== LS844s- 1 = '113.1 rpm

12.3665ms- 1

Ps == 21298kW

= Pv

24.041k

=

Ps 77s

=

21298 x 0.970 = 20659 k W

T == 150lkN

PE = 15254 kW (from PE

RT = PE = F Pv J(QB

=

15254

12.3665

2... pn 3 D5

==

=

V curve)

-

1233kN

== 211' x 1.025

20659 X 1.88443 x 7.0 5

=

0.02852

1501 = 0.1718 1.025 x 1.88442 x 7.0 4

Torque identity: 1(Q

=

110

==

,~

wQ

1(QB 1(T 1(Q

=

0.02852

J == 0.7209

1(T

= 0.1686

J 0.1686 0.7209 == x- == 0.6783 271" 0.02852 271"

JnD

== 0.7209 x 1.8844 x 7.0 == 9.5092ms- 1

== 1- -'-A V RT

t == 1 - -

T

=

1_~0~ == 0.2311

=

1233 1 - - - = 0.1785 1501

12.3665

Ship THals and Service Performance

293

0.8215

I-t

1.0684 =0.7689 =

7]H

=

--

7]R

=

KTB

7]D

= 7]0 7]R T/H = 0.6783 x 1.0190 x 1.0684 = 0.7385

l-w

0.1718

= 1.0190 0.1686

=

KT

,.

= 15254 = 0.7384 20659

Thrust identity: J

KT J

7]0

= KQ 211"

VA

=

WT

= 1- VA V

t

= 0.1785

7]H

=

7]R

=

\

7]D

=

=

I-t

!(Q KQB

!

= 0.6748

JnD = 0.7140 x 1.8844 x 7.0

l-w

KQ = 0.02893

0.7140

9.4182 = 1- 12.3665

= 0.2384

=

0.8215 0.7616

=

1.0787

=

0.02893 0.02852

=

1.0144

7]0 7]R 7]H

=

0.6748

X

- 9.4182ms- 1

1.0144

X

1.0787

= 0.7384

As this example shows, if the propeller thrust is measured during the trials, it is possible to analyse the results by both thrust identity and torque identity, and to obtain the individual components of the propulsive efficiency. It is, however, necessary to have an estimated value of the effective power based on model tests and a suitable correlation allowance. Alternatively, one may assume that the propulsion factors based on mod~l tests after suitable corrections for scale effects apply to the ship, on trials, and determine the correlation allowance on effective power determined from model tests.

294

10.4


Bollard Pull Trials

Bollard pull trials are carried out to determine the maximum static pull that can be exerted by a vessel used for towing duties, e.g. a tug. The tug is attached by a long towline to a bollard on the quayside. A load cell or other device to measure the tension in the towline is inserted at the bollard end of the towline. The propeller is run slowly at first until the towline becomes taut and then quickly brought up to the maximum rpm that can be achieved without exceeding the specified or the maximum permissible torque or rpm of the engine. The bollard pull (i.e. the tension in the towline indicated by the load cell), the propeller rpm and the engine power are measured. If necessary, special instruments may be fitted to the propeller shafting to measure rpm and power. The bollard pull rises sharply as the propeller rpm is increased initially and becomes steady when the rpm becomes constant, although there. is usually· a small cyclic variation caused by oscillations in the towline and by rudder movements needed to keep the vessel in a fixed position. After some time however, the disturbance imparted to the water by the propeller is reflected back to the vessel by the quay walls, and this usually causes the bollard pull to fall. It is therefore important to have a sufficiently long towline (at least two ship lengths) and to position the vessel with respect to the quay walls in such a way as to delay and minimise the reflection of the disturbance as much as possible. The vessel should also be well clear of the shore and the depth of water should be sufficient (at \ least twice the draught aft). As far as possible, bollard pull trials should be carried out in a location such that the effects of currents, tides, waves and win'd are minimal. It is also desirable to have sufficient space to al.low for the slight sideways oscillation of the vessel that often occurs during the bollard pull trial, and to allow the vessel to stop safely in case the towline parts. I

10.5

Service Performance Analysis

The propulsive performance of a ship at sea is determined by the relation between the ship speed, the engine power and the propeller revolution rate. The relationship depends basically upon three parameters: (i) the displace-' ment and trim of the ship, (ii) the environmental conditions, and (iii) the condition of the hull surface and the propeller. It is therefore necessary to

t

I


295

measure and record the speed, power and propeller rpm together with values of displacement and trim, measures representing the environmental condi tions (wind and waves) and the condition of the hull and propeller of the ship in service. The data so collected can be analysed to find the effect of displacement and trim, wind and waves, and hull and propeller condition on the speed-power-rpm relationship. The results of such service performance analyses 'can then be used to devise methods to operate ships in an optimum· manner. Unfortunately, there are many difficulties in obtaining reliable service per formance data unless special instrumentation and specially trained personnel ar~ provided on board the ship. The development of automatic data logging equipment. has greatly improved the reliability, of the data on engine power andprop,eller rpm, but the measurement of snip speed is still problematic. The speed is usually determined by observatidn of the ship's position at reg ular intervals, e.g. once every watch. This speed must then be corrected for the currents in the water on the basis of oc~an current data. Alternatively, one may determine ship speed by using a ,speed log but this depends upon the calibration of the log, which changes with time. A third method is to use what is known as the Dutchman's login which objects are dropped into the sea sufficiently far from the ship and the time taken, for them to float a known distance past the ship determined. In modern times, one may use satellite global positioning systems to determine ship speed. The displacement and trim of the ship can be determined accurately at the start and the end of a voyage, and a record of the consumption of fuel, water, provisions etc. provides reasonably accurate estimates of displacement and trim during the voyage. The condition of the hull surface and the propeller can only be determined when the ship is dry-docked, and the average hull roughness and the roughness of the propeller surface measured. It is, how ever, more convenient to represent the condition of the hull and propeller in terms of the "number of days out of dry-dock" for the purpose of service performance analysis. This simple criterion ignores the effect of such factors as the number of days in port and at sea, and the number of days in tropical waters and in temperate waters, which affect the rate of fouling and hence the condition of the hull surface. However, for a ship following a more or less fixed route and voyage schedule, days out of dry-dock is a sufficiently accurate measure of hull and propeller condition.

296


Numerical measures of the environmental conditions in which a ship op erates are also necessary for the analysis of service performance. The ",,;nd speed and direction relative to the ship are measured by an anemometer and wind vane on board, and the true wind velocity then determined. The average wave height and the wave direction relative to the ship are usually estimated by the ship staff. The effect of wind on speed and power can be determined with the help of model tests in a wind tunnel. Similarly, model tests in waves can be used to estimate the effect of waves on speed and power. Such model test data are not always available, and normally much simpler methods are used for service performance analysis. The true wind speed is related to the Beaufort number, which is widely used as a measure of the environmental conditions, and this together with a wave direction factor may be used to obtain a "weather number" for the purpose of analysis. Service performance data consisting of observations of power, speed and rpm and values of displacement and trim together with days out of dry . dock and estimated wind and wave conditions must then be analysed. The

object of the analysis is to separate out the effects of the three major' factors:

displacement, weather and days out of dry-dock. There are two approaches

to the problem, the first involving heuristic methods and the other using

statistical methods. ' In the heuristic methods, the data are first corrected to a standard dis placement and speed. This may be done using model test data for different displacements and trims, but more usually it is assumed that the power is rel~ted to displacement and speed as follows:

(lOA) Other formulas for correcting power to a standard displacement are:

PSI

Pso

=

Al 0.65 .60

+ 0.35

(10.5)

0

and: PSI P so

=

\lI/ S l \10 / So

(10.6)


297

where Ps is the shaft power, f::!,. the displacement, 'V the displacement volume and S the wetted surface, the subscripts 0 and 1 referring to the standard and any other displacement respectively. The effect of displacement on the power may also be obtained directly from the service performance data by plotting a power coefficient such as f::!,. 2/3 V 3 / Ps as a function of displacement and drawing a trend line. It is preferable to consider the loaded and the ballast voyages separately since these 'simple methods may not be able to cater to large changes in displacement and trim. The effect of weather must then be determined. For this purpose, it is necessary to devise a numerical measure of the weather conditions for the period during which power, ~peed and rpm are measured. A simple method is to use the Beaufort number (based on the true wind speed) together with a wind direction factor. Alternatively, the power corrected to a standard displace~ent and speed can be plotted as a function of the Beaufort number for different wind directions, viz. ahead, bow quarter, stern quarter and astern.. Another method is to classify the weather encountered into groups, assign a numerical value to each group, ~nd obtain a weighted average of the weather number for each voyage. Burrill (1960), for example, divides the weather into four groups-good (0-25), moderate (25-50), heavy (50 75) and very heavy (75-100), and derives a "weather intensity factor":

W

:;

,.

1: ~Wi = 1: di

(10.7)

where W is the weather intensity factor and ~ the number of days in which the weather had a numerical value Wi (12.5 for good, 37.5 for moderate, 62.5 for heavy and 87.5 for very heavy weather) in a particular voyage. The average value of the Admiralty Coefficient f:j.. 2/3 V 3 / P s for each voyage is then plotted as a function of W, and the mean curve through the points then gives the effect of weather on the ship's performance. Once the effect of weather on the power at a given displacement and speed has been determined by one of the methods described in the foregoing, the power can be corrected to a specific weather condition, e.g. to still air and calm sea conditions, or perhaps "moderate" weather. Finally, the power corrected to a standard displacement and speed and to a specific weather condition can be plotted as a function of days out of dry-dock to determine the effect of hull and propeller condition on power.

298


When this is done over several dry-docking cycles, it is observed that the cleaning and painting of the hull does not bring the performance of the ship back to the level at the .end of the previous dry-docking. Thus, the effect of time in service on the performance of a ship can be divided into two components, the effect of fouling of the hull surface which is removed when the ship is dry-docked, and the effect of permanent hull deterioration which cannot be eliminated by cleaning and painting of the hull surface and which grows slowly but steadily as the ship ages.

Ps/n 3

One may also calculate the values of and V/n for each set of mea surements and then determine the wake fraction based on torque identity with the help of the open water characteristics pf the propeller. If the values of power are corrected to standard displacement, speed and weather con ditions, the effect of hull surface conditions on the wake fraction may be determined. Example 5 A ship of design displacement 19QOO tonnes and speed 16.5 k has an engine of 9500kWat 130rpm directly connected to a propel~er of diameter 5.5m l:\.nd pitch ratio 0.75. The open water characteristics of the propeller are as follows: .

J

0

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

KT

0.328

0.301

0.270

0.237

0.200

0.160

0.116

0.070

0.020

10KQ:

0.366

0.346

0.320

0.290

0.255

0.215

0.170

0.120

0.065

'.

The ~ollowing data are collected for the first two years of service during which the ship made 25 voyages and was dry-docked at the end of 383 days for a period of 15 days for cleaning and painting of the hull and routine maintenance. Voyage Averages

Voyage No. 1 2 3 4 5

Days in Displacement Speed Shaft Power Propeller Weather

Service l:!. Rpm

V Ps tonnes knots kW n

do 15 45 73 101 133

18100 18400 17500 18200 18100

16.7 16.4 16.3 16.0 15.8

6680 7090 6750 7360 8050

130.1 130.9 129.1 131.2 133.3

Good Moderate Moderate Heavy Very Heavy

\

I


299

Voyage Averages Voyage No.

6 7 8 9 10 11

12 13 14 15 16 17 18 19 20 21 22 23 24 25

I

Days in Displacement Speed Shaft Power Propeller Weather Service V Ps A Rpm knots n tonnes kW do

164 194 223 253 284 313 342 369 416 448 478 508 537 567 595 628 656 685 713 744

16.4 15.9 16.1 l6.0 15.7 15.7 16.1 16.0 16.6 16.1 16.0 15.8 16.2 16.1 15.8 15.6 15.9 15.9 16.0 15.4

18800 17900 17600 18200 18300 18600 17900 18100 18400 17400 18600 18000 17800 18200 18800 17700 17600 18200 18300 17900

6750 7360 6810 6850 7310 7460 6340 6290 6790 7410 7720 8250 7010 ,,'7060 I 7660 , 8160 6780 7020 6480 8110

128.5 129.8 127.8 127.8 129.2 129.6 125.1 124.7 129.8 130.9 131.6 133.7 128.5 128.6 130.3 132.2 126.6 127.2 125.1 131.2

Good Heavy Moderate Moderate Heavy Heavy Good Good Good Heavy Heavy Very Heavy Moderate Moderate Heavy Very Heavy Moderate Moderate Good Heavy

Analyse these data.

\

A method of analysing these data is given in the following table: Voyage No. 1 2 3 4 5 6 7 8 9

Days out of Weather 63y3 6,iy3 P p. Dry-dock Number -----p; s d1 W kW

s

15 45 73 101 133 164 194 223 253

12.5 37.5 37.5 62.5 87.5 12.5 62.5 37.5 37.5

481 434 432 385 338 462 374 415 414

1

~

__

-~----.----_.

.-..

436 434 432 428 424 419 417 415 414

6476 6512 6529 6591 6652 6744 6770 6810 6825

Y nD 0.7203 0.7031 0.7085 0.6843 0.6651 0.7162 0.6874 0.7069 0.7026

10KQB

J

WQ

0.1961 0.2043 0.2028 0.2107 0.2197 0.2056 0.2176 0.2109 0.2121

0.5405 0.5198 0.5237 0.5036 0.4800 0.5165 0.4857 0.5030 0.4998

0.2496 0.2607 0.2608 0.2641 0.27,83 0.2789 0.2935 0.2885 0.2886

300


Voyage No. 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

Notes:

3 Days out of Weather b.i V Dry-dock Number Ps

d1

IV

284 313 342 369 18 50 80 110 139 169 197 230 258 287 315 346

62.5 62.5 12.5 12.5 12.5 62.5 62.5 87.5 37.5 37.5 62.5 87.5 37.5 37.5 12.5 87.5

b.~ V 3

P's

V nD

10KQB

J

WQ

0.6819 0.6798 0.7222 0.7200 0.7177 0.6902 0.6823 0.6632 0.7075 0.7025 0.6805 0.6622 0.7048 0.7015 0.7177 0.6587

0.2121 0.2215 0.2093 0.2097 0.2007 0.2136 0.2190 0.2231 0.2136 0.2146 0.2238 0.2283 0.2160 0.2205 0.2139 0.2321

0.4816 0.4752 0.5070 0.5061 0.5290 0.4961 0.4820 0.4709 0.4961 0.4934 0.4691 0.4572 0.4898 0.4779 0.4951 0.4468

0.2937 0.3010 0.2980 0.2971 0.2629 0.2812 0.2936' 0.2898 0.2988 0.2977 0.3106 0.3096 0.3051 0.3186 0.3102 0,3216

Ps kW

368 364 450 449 470 378 372 328 413 409 364 316 401 . 396 . 439 308

411 408 407 406 426 422 416 415 413 409 407 403 401 396 396 395

6870 6929 6937 6962 6626 6699 6791 6803 6829 6904 6931 7012 7039 7127 7138 7151

...

Days out of dry-dock d1 = . do for the first 13 voyages = do - 398 for the remaining 12 voyages Weather Number

n-

= 12.5 for good, 37.5 for moderate, 62.5 for heavy and 87.5 for very heavy weather,

Correytion for weather according to: A~ V 3

~

=

A~ V -p;-

AS V 3

----p;- + 1. 7862 ( W

- 37.5 )

3

obtained from a plot of

as a function of W.

Shaft power corrected to moderate weather and to 18100 tonnes displacement and 16.0 knots speed (approximate average values for the 25 voyages) given by:

Pi; = V nD

18100 2 / 3 x 16.03 (.06. 2 / 3 V 3 /Ps)

V in m per sec

Ship THaIs and Service Performance

=

I

I

1]s

= 0.970 assumed.

from propeller open water characteristics

J

WQ

PS1]S

10 21f'pn3 Dr-' "

301

=

J 1- WinD)

These data can be plotted in various ways. A graph of 6,2/3 V 3 I Ps as a function of the weather number W shows the effect of weather on power, andc'an be used to obtain 6,2/3 V 3 IPs where P is the power for a specific weather condition, "moderate" in the example. The power for a specific displacement, speed and weather condition is then determined, and plotted as a function of days in service to show the effect of hull surface conditions. Finally, the torque coefficient KQ is calculated and used to find the torque identity wake fractionwQ, which is plotted as function of days in service. One must expect such diagrams to show a fair amount of scatter.

s

In statistical methods of analysing servic~ performance data, linear multi ple regression analysis is used to fit an equation of the form:

(10.8)

to the data such that 2:( Yest - y)2 is minimised. The dependent variable Y may be the shaft power Ps or a power coefficient such as Psi 6,2/3 V 3 and the independent variables Xl, X2, ••• may be displacement, trim, speed, a weather number, days out of dry dock, or combinations of these quantities and other parameters. The variables included in a regression analysis must be carefully chosen, and the significance of each variable must be examined by determining the magnitude of the reduction in 2:( Yest - y)2 caused by the inclusion of that variable. For further details, one may refer to a book on Statistics.

Problems 1.

A ship with a propeller of 4.0 m diameter and 0.8 pitch ratio is moored along the quayside by two mooring ropes. During dock trials, the propeller is run at 90 rpm and the mooring ropes become inclined at 15 degrees to the ship


302

centre line. Calculate the power delivered to the propeller, the tension in the mooring ropes and the' reaction between the ship and the quayside. Assume that the thrust deduction fraction is 0.050, the relative rotative efficiency

is 1.000 and the open water characteristics of the propeller are as given in Table 4.3.

2. A ship on speed trials makes six groups of runs, each group consisting of two runs in opposite directions at'the same power setting over the measured mile. The following data are recorded: Run No.

Direction

Time at start

Time over mile

Shaft power

hr-min

s

kW

Propeller revolution rate rpm

8-51 9-38 10-25 11-08 11-52 12-32 13-12 13-49 14-24 14-59 15-32 16-04

321.8 455.1 299.4 369.9 293,1 315,6 287,5 267,3 273.1 238.1 249.0 214.6

1481 1501 1992 2020 2525 2585 3320 3360 4182 4242 5095 5155

87.8 88.4 97.5 98.3 106.6 106.2 116.2 117.2 125.7 126.1 136.5 137.1

Propeller thrust

.,

tl.

~

~ f~ ~

~

~f;1

~.l

}~~

~. t)~_

~\'i

~.~ ~,

kN

:.1~l

241 247 291 . 295 334 352 402 410 459 465

522

528

lff

'4.:'

1 2 3 4 5 6 7 8 9 10 11 12

N S N S N S N S N S N S

Determine the speed of the ship through the water for each run and plot the speed of the current to the North as a function of the time of day. Calculate also the average speed, shaft power, propeller rpm and thrust for each of the six groups of rWlS. 3. The effective power of the ship in Problem 2 is as follows: Speed, knots

9.0

10.0

11.0

12.0

13,0

14,0

15,0

16,0

Effective power:

856.

1084

1404

1792

2225

2679

3130

3554

The propeller of the ship has a diameter of 5,0 m and a pitch ratio of 0.8, and

its open water characteristics are as given in Table 4.3. Analyse the data of

the trials by both thrust identity and torque identity. The shafting efficiency

is 0.970.

~ .:r

~l~:

_"{t_ ~-.;.

<

1\

~~ ~I,

i{: ~

.~

q

ij


303

A ship is taken out on trials and the following observations recorded:

4.

Run Direction Time Time Average relative wind Direction :\0. at start on mile Velocity

Shaft power

hr-min

secs

mpersec

deg

kW

Propeller revolution rate rpm

1 2 3

N S N

8-04 8-46 9-28

229.1 265.7 228.1

11.3 20.0 10.3

23.5 167.4 23.2

2350 2300 2360

73.6 72.8 73.1

4 5.

S N S

10-16 10-52 11-40

236.9 210.2 219.9

22.3 10.9 22.6

179.8 15.2 175.3

3600 3640 3610

83.4 83.8 83.5

7 8 9

N 'S

N

12-16 12-58 13-39

182.2 218.8 180.1

12.0 24.7 12.1

4.1 0.5 170.5

7360 7370 7380

104.6 104.2 104.5

10 11 12 13

S N S N

142-2 15-05 . 15-40 16-17

194.6 171.6 189.3 176.5

26.3 14.6 26.3 15.2

7.7 161.4 11.6 157.1

9760 9860 9830 9790

113.7 114.1 114.9 114.0

6

The ship has an equivalent above-water transverse projected area of 250 m 2 , and its effective power based on model tests with a correlation allowance of 0.4 x 103 is as follows: \

Speed, knots

13.0

15.0

17.0

19.0

21.0

Effective power, kW:

1477

2220

3439

5135

7306

Determine the speed of the ship corrected for wind and current, the shaft power and the propeller rpm for each of the four groups of runs. Determine also how the speed of the current varies with the time of day. Use Eqns. (10.2) . and (10.3) to estimate the wind resistance. 5.

The ship in Problem4 has a wetted surface of 3750m2 • Its propeller is of 6.0 m diameter and 1.0 pitch ratio, its open water characteristics being as given in Table 4.3. The shafting efficiency is 0.970, and the propulsion factors determined from a self-propulsion test on the basis of torque identity are as follows:

V knots: wQ

L

13.0 0.2255

15.0 0.2349

17.0, 0.2392

19.0 0.2384

21.0 0.2327

304

Basic Ship Propulsion t TJR

0.1757 1.0183

0.1784 1.0245

0.1797 1.0274

0.1796 1.0271

0.1781 1.0236

Determine the correlation allowance and the wake scale effect.

m 1.

6. A ship has an engine of maximum continuous rating 20000 kW at 114 rpm. The equivalent above water transverse projected area of the ship is 300 m2 • The contract for the ship requires it to have a minimum speed of 18 knots in calm water at 85 percent maximum continuous rated power and 100 percent rated rpm. During the acceptance trials, the following data are recorded: Run No. 1 Direction W Time at start, hr-min 10-30 Time on mile 207.4 Relative wind velocity, m per sec: 9.908 Wind direction off the -bow, deg 7.5 Shaft power 16920 113.5 Propeller rpm

2 4 5 3 W E W E 10-57 11-25 11-53 12-19 ! 190.5 202.6 201.8 187.9 7.495 11.164 7.765 10.468 17.0 13.9 14.3 9.1 16930 16980 16960 16950 113.6 114.2 114.1 114.0

The slope of the effective power-speed curve of the ship at 18 knots is 2300 kW per knot. The wind resistance of the ship may be estimated using Eqns. (1O.2) and (10.3). Determine the speed that the ship would attain in calm water, still air conditions. Does the ship fulfil the contract requirements? 7. A single screw ship has an engine with a maximum continuous rating, when new, of 10500 k\V at 126 rpm directly connected to the propeller of diameter 6.2m and pitch ratio 0.7. The effective power of the ship in average service conditions when new is as follows: Speed, knots Effective power, kW:

15.0 3699

15.5 4149

16.0 4637

16.5 5164

17.0 5733

17.5 6345

The wake fraction is 0.250, the thrust deduction fraction 0.200 and the relative rotati\;e efficiency 1.030 based on thrust identity. The shafting efficiency is 0.970. As the ship ages, its effective power in average service conditions increases uniformly at the rate of 5 percent per year, so that at the end of 10 years the effective power is 1.50 times the effective power when the ship was new. The wake fraction similarly increases at the rate of 0.5 percent per year and the thrust deduction fraction at the rate of 0.7 percent per year. The changes in the relative rotative efficiency and the shafting efficiency are' negligible. The maximum continuous power rating of the engine decreases by

----------------

--------- ----


305

1.0 percent every year. Determine the changes in the speed and power of the ship in average service conditions at the end of every year for ten years. If the propeller is to be changed when the power required exceeds the maximum available power at 126 rpm, in which year should the propeller be changed? What should be the pitch ratio of the new propeller if it is to have the same diameter as the original propeller and run at 126 rpm with 90 percent of the available power? What will be the resulting ship speed? Use Table 4.3 for the propeller open water characteristics. 8. The service performance data of a ship are collect,ed for a two year period during which the ship made twenty voyages. The average displacement of the ,ship during these voyages was 15000 tonnes and the average speed 16.0 knots. In the following data, the brake power has been corrected to correspond to the average displacement and speed, and the cle~n hull condition. Voyage No.

Brake power kW

Wind speed knots

Wind direction deg

1

7370 7570 11520 9290 7120

2 2

100 50 35 15

2

3 4 5 6

7 8 9

10 11 12 13

14 15 16 17 18 19 20

23

14 9 24

140

10310 8160 9570 10190 7640 7770 8480 8690

19 9 30 26 20 15 9

10

9365

18

6990 7480

10

8360

28

8850

25 5

8140

7930

~2

2

165 85 125 75

150 130 40 70 60 175 115 180 95 25

(True wind speed and direction off the bow) The brake power in calm water for 15000 tonnes'displacement and 16.0 knots speed is 7500 k\V. Analyse these data to obtain curves showing the percentage

i_

306

Basic Ship Propulsion increase in power over that in. calm water as a function of the Beaufort number for (a) head seas (0-45 degrees off the bow), (b) bow quartering seas (45-90 degrees off the bow), (c) stern quartering seas (90-135 degrees off the bow), and (d) follo\\;ng seas (135-180 degrees off the bow). The relation between wind speed and Beaufort number is as follows: Beaufort No.

Wind speed knots

Beaufort No.

Wind speed knots

1 2 3 4

2

5 6 7 8

18-20 22-26 28-32 34-40

5

8.5-10 12-16

Hence, find the percentage increase in power for a wind speed of 5 knots for the four direction ranges. .

9.

A ship has a design displacement of 13000 tonnes and a speed of 17.0 knots. In its first four years of service, it makes eleven voyages each year, and is then dry-docked when its hull is cleaned and repainted. Average values of speed and power observed during voyages when fair weather prevailed are given in the following table: Voyage No.

1 5 11

Mean days out of drydock 10

120 310

Displacement

Average speed

Average power

tonnes

knots

kW

11000 12000 9000

17.0 16.8 17.2

8586 9029 8218

13000 9500 10500

16.8 17.1 17.1

9450 8333 9069

12500 11500 9750

16.7 16.8 17.2

9166 9099 8941

Ship dry-docked

12 18 21

25 170 300

Ship dry-docked

23 28 33

20 160 320

Ship dry-docked

i

Ship Trials and Service Performance Voyage No.

34 39 43

Mean days out of drydock 15 140 290

307

Displacement

Average speed

Average power

tonnes

knots

kW

12750 10250 9250

16.7 17.1 17.3

9417 8997 8886

Correct these data to the design speed and displacement and plot the cor rected power to a base of days in service. Attempt to separate the effect of fouling which is removed at each dry-docking from the effect of irreversible hull deterioration. Hence, predict the power required at the end of the fifth year in service just before the ship is dry-docked. 10.· A ship has a propulsion plant consisting of a two-stroke turbo-charged diesel engine of maximum continuous rating 10500 kW at 144 rpm directly connected to a propeller of 5.0 m diameter and 1.0 pitch ratio. The propulsion factors based on thrust identity are: wake fraction 0.240, thrust deduction fraction 0.200, relative rotative efficiency 1.030. ['he shafting ~fficiency is 0.980. The effective power of the ship in calm water is as follows: Speed, knots Effe~tive power, kW:

\

15.5 3926

16.0 4402

16.5 4918

17.0 5475

17.5 6078

18.0

6726

It is desired to estimate the effect of weather on the service performance of the ship. For this purpose, draw two sets of curves showing (a) brake power as a function of engine rpm, and (b) ship speed as a function of engine rpm for increases in effective power of 0, 10, 20, 30, 40 and 50 percent over that in calm water. Hence deduce the resulting drop in ship speed and engine rpm as bad weather causes the given increases in effective power. Assume . that the engine torque and rpm at the maximum continuous rating cannot be exceeded.

CHAPTER

11

Some Miscellaneous Topics 11.1

Unsteady Propeller Loading

The thrust and torque of a propeller operating behind a ship are not con stant even when the ship is moving at a constant speed in calm water with the propeller running at a fixed rpm. Fluctuations in the thrust and torque arise because the flow into the propeller varies along the circumference at each radius. This results in a periodic variation of the relative velocity of the flow with respect to a point on the propeller, and produces an unsteady loading on the propeller blades. The fundamental frequency of these un steady propeller forces is the propeller revolution rate or "shaft frequency" but higher harmonics also exist. Unsteady propeller loading causes several adverse effects. The periodic forces ,and moments created by the propeller operating in a nOll-uniform flow are tr~nsmitted through the propeller shaft bearings and through the water to the hull of the ship, leading to vibrati~'iJ. of the propulsion shafting system and the hull. The periodic stresses in the propeller, the shafting and the hull due to unsteady propeller loading may eventually result in fatigue failure. Non-uniform flow into the propeller may also result in periodic cavitation causing enhanced vibration, noise and erosion. The severity and importance of these adverse effects vary with the type of vessel. Some hull vibration clue to periodi.c cavitation is fairly common in merchant ships with single propellers operating with the minimum permissible clearances between hull and propeller, and is usually acceptable. In warships, on the other hand, 308

309

Some ;Afiscellaneous Topics

noise due to periodic cavitation may be unacceptable. Fortunately,. high speed warships have twin screws with comparatively large clearances from the hull, so that the flow into the propellers is much more uniform than in a single screw merchant ship, and the effects of unsteady propeller loading are small. r

r

'R=0.7

R .. e.7 1.0 .------.."..---,.-.."..----,

1.0 . - - - - - - - - , - - - - - - ,

0.5 t - r - - - - H - t - - - - ' r - i

0.5 !L-\:----"7"---.,e>;.;;;-----T7f

Va

Vt

V

o '-, 0

V ----l.

e degrees

--1

180360

-0

I-.------'--~-----'

0

e degrees

180

.360

Figure 11.1: Circumferential Variation of Velocities in the Propeller Disc. i

The axial and tangential components of the flow velocity into the propeller disc of a single screw ship typically vary around the circumference at a given radius as shown in Fig. ILL Va and vt are the axial and tangential velocities, e is the angle measured from the vertical upward, and V the ship speed. Owing to the port and starboard symmetry of the ship hull, the axial velocity should be an even function of e, i.e. Va(e) = Va(-e), and the tangential velocity an odd function, i.e. vt(e) = -vt( -e). This is not always found to be so because of experimental errors in the measurement of these velocities and also because of vortex shedding by the hull in full form ships. For a propeller not on the ship centre line, there is naturally no flow symmetry. In general, therefore, the axial and tangential velocities in the propeller disc can be represented in terms of Fourier Series:

\

00

Va (r, e)

I: {am(r) cosme + bm(r) sin me}

m=O

00

vt(r,e) = I:{a~n(r)cosme+b~(r)sinme} m=O

L

(11.1)

310


Trun.cating these series at m representation.

= 15 or 20 usually gives a reasonably accurate

The Fourier series representation of the axial and tangential velocities may be used to determine the unsteady propeller forces. Various methods for doing so exist, of which the simplest is based on a quasi-static approach. The thrust and torque of t.he i th blade at an angle 0 to the upward vertical are given by:

T:.

fl.O dT (Xl 0) dx

=

dx

JXb

_1l.

0 dQ (X, 0) Qi - . d dx Xb

.

(11.2)

X

where dT(x, 0) and dQ(x, 0) are the thrust and torque of a blade element dx at x = 1) R calculated for the known velocities Va(x,O) and vt(x,O). Xb is the non-dimensional boss radius. The thrust and torque for all the !Z blades will then be:

z

T(O) =

Q(8)

=

.

~T:.

t.

[0"+ 27r(~-1)]

Q,

[0+ 2~( ~-1 1]

(11.3)

and the mean thrust and torque of the propeller will be:

T =

~ f27r T(e) de 27r

Jo

(11.4) Q

= 227r

f27r Q(O) de

Jo

The tangential force on the i th blade of the propeller at the angle by:

e is given

Some Miscellaneous Topics

311

1

1.0

i

F

; I

-

Xb

_1_ dQ(x, 8) d X R dx x

(11.5)

so that for all the Z blades, the tangential force and its horizontal and vertical components are given by:

\'

r'

I

r

I:

F(O) = t,Fi [0 + 2~(iZ-

t,

FH(O)

F,,(O)

~

F,

1)]

[0+ 2~( Z- 1)] i

t,Fi [0 +

2~( i Z

cos [0

+ 2~( i Z

1)]

(11.6)

1)] sin [0 + 2~( iZ-l )]

It is assumed that unsteady propeller loading is a linear phenomenon, i.e. a pure harmonic velocity variation of a given frequency will produce a pure harmonic force variation of the same frequency with possibly a phase differ ence. Also, the resultant of several harmonics is the linear superposition of those harmonics. If the velocity components are represented by Eqn. (11.1), the resulting thrust and torque of the i th blade are represented by:

00

Ti(O) =

L

{AmTcosmO

+ B m Tsinm8}

m=O

(11.7)

00

Qi(O) =

L {A

mQ

cosmO

+ BmQ sinm8}

m=O

!

I

L

The thrust and torque for all the Z blades' will be:

312


+ BmT sin [ me + 211" ( i ;=--1'--)m_] } (11.8)

. [ + B mQ sm me + 211" ( i -Z I )

m] }'

It can be shown that:

'\.~ [e 2" ( i -Z I ) m] L...Jcos m + i""l

"Z' . .

[

L...Jsm m

=

Z c;osme

=

0

for m

otherwise

e+ 2;; ( i -Z I )

=

kZ,

k

0,1,2,3 ...

(11.9)

m]

.i

i=;~

= 0

for all m i,

Therefore, owing to the angular symmetry of the propeller (Z identi cal blades spaced 2rr/Z radians apart), the only harmonics in the veloc ity field that result in a net thrust and torque are those corresponding to m = 0, Z, 2Z, 3Z .... The propeller thus filters out many of the frequencies present in the velocity field. The direction of the tangential force Fi (e) on the i th blade changes with e, and it is therefore necessary to consider its horizontal and vertical com pOl1€nts.,.w.hose directions are fixed. The horizontal component for all the Z blades in the velocity field given by Eqn. (11.1) may be represented as: '

313

Some A-fiscellaneous Topics

(8 +

211"(i-l)m)} Z cos

(8 + 211"(i-l))]

+

. m B mH sm

+

cos[(m-l)e+211"(~-I)(im-l)]}]

Z

,.,f' I

i

t'

i i

Z + ~L 00

t=l m=O

+

[1

{[ ! 2(' - 1) ] sin (m+l)8'+ 11" ~ (m+l)

ZBmH. '

sin [( m - 1 ) 8 + 211" ( ~:1 ) ( m - 1 )] } ]

(11.10) The various terms in Eqn. (11.10) all add up to zero when summed up over the Z blades except whenm + 1 = kZ or m - 1 = kZ, k = 0,1,2, .... A similar result can be obtained for the vertical component Fv(8) of the tangential force. Thus, the tangential force F(8) is excited only by those harmonic components in the velocity field for which (m+ 1)/Z or (m -1)/Z are integers. For other values of m, the individual contributions of the Z blades cancel each other. The calculation of the periodic forces on the propeller as a function of the angular position 8 from the experimentally determined propeller charac teristics using a quasi-static approach is simple. However, this method has been found to give variations in thrust and torque which are nearly twice the values observed in model experiments. More sophisticated methods are therefore necessary. An improvement on the quasi-static approach consists in determining the thrust and torque on the blade elements at various radii, dT(x,8) and dQ(x,8), using two-dimensional unsteady aerofpil theory. An even more accurate method for determining unsteady propeller loading is based on using unsteady lifting surface theory.

314


Theoretical and experimental studies of unsteady propeller loading confirm that the fluctuations in the thrust and torque of all the blades arise only from those harmonics in the velocity field that are an integer times the number of blades, (m = kZ, k = 0,1,2,3, ...). The tangential force and the resulting moments at the propeller shaft bearings arise only for those harmonics for which m + 1 = kZ or m - 1 = kZ. The harmonics in the velocity field at the propeller depend mainly on the shape of the afterbody of the ship. The amplitudes am and bm in Eqn. (11.1) decrease with increasing m, this decrease being monotonic for open sterns in single screw ship while for conventional sterns the amplitudes are greater for m = 0,2,4, ... than for m = 1,3,5,. '" The tangential velocity variation is usually large and has

a strong first harmonic.

~\

In the design of propellers to minimise unsteady propeller loading for a given non-uniform velocity field, calculations are carried out for different values of number of blades and different skew magnitudes and radial distri butions. That combination of blade number, skew magnitude and its radial distribution is selected which gives the minimum fluctuations in thrust and torque and the smallest values of the tangential force components and the resulting moments. Excessive magnitudes and extreme distributions of skew may not be acceptable becau.se they would cause difficulties in propeller man ufacture. In some cases, it may be necessary to consider modifications to the afterbody of the ship to restrict unsteady propeller loadings to acceptable levels. \

Example 1 !

A single screw ship has a four-bladed propeller of 5.0 m diameter and 0.8 pitch ratio running at 150 rpm with the ship moving ahead at a speed of 10.85 m per sec. The axial and tangential components of the relative velocity of water with respect to the ship in the plane of the propeller at 0.7 R are given in m per sec by: %(8) = 8.0000 - 0.4 cos 8 - 0.3cos2B + 0.2cos38 - 0.1 cos 48 + 0.05cos58

Vt (8) ::: -1.2 sin 8 + 0.6 sin 28 - 0.3 sin 38 + 0.15 sin 48 - 0.075 sin 58 Determine the values of the thrust, the torque and the horizontal and vertical

components of the tangential force for different values of the blade angle 8. Assume

that the blade section at 0.7R represents the whole propeller, and that the centroid

of the tangential force on each blade is at a radius of 1.5 m. The propeller open

water characteristics are given by:

.. J ~.: l

...

~

!

1 .


315

KT = 0.3415 - 0.2588 J - 0.1657 J2 10KQ = 0.4021 - 0.2330 J - 0.2100 J2 Z

=4

D

= 5.0m

P D

= 0.8

n = 150rpm ::: 2.58- 1

Centroid of tangential force on each blade at a radius

TQ

V

= 10.85ms- 1

= 1.500 m

For a. blade at an angle B to the upward vertical, the tangential velocity relative to the water is given at 0.7R by:

The instantaneous revolution rate at the angle B is then:

n(B) = Vrt(B) O.71rR

=

0.71fRn - Vt(B) ::: n _ "t(B)

O.71rR 0.71fR

The advance coefficient for a blade at "the angle B is:

J(B) =

Va(B)

n(B) D

for which the thrust and torque coefficients can be obtained from the open water characteristics. The thrust and torque of the blade at the angle B are ~hen: \

T1(B) :::: Q1(B)

~KT(B) pn2 (B) D 4

= ~KQ(B) pn2 (B) D S

The tangential force and its horizontal and vertical components are: F1(B) = QI(B)/rQ

FlH(B) = Fl(B)cosB

F l v (B) = F I (B) sin B

The calculations are given in Tables 11.1 (a), (b), (c) and (d).

, I

L


318 Table 11.1 (c) (Contd.)

e

T 1 (e)

T2 ( 9)

T 3 (9)

T4 (e)

T(8)

deg

kN

kN

kN

kN

kN

75 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 315 330 345 360

138.74 138.68 137.06 139.65 150.56 160.75 151.67 117.46 74.19 42.54 32.13 39.23 52.81 65.80 78.77 93.11 106.25 115.40 121.91 128.52

52.81 . 121.91 151.67 65.80 128.52 117.46 78.77 134.63 74..19 42.54 93.11 137.53 32.13 106.25 137.41 39.23 115.40 137.47 52.81 121.91 138.74 65.80 128.52 138.68 78.77 '134.63 137.06 93.11 137.53 139.65 106.25 '137.41 ' 150.56 115.40 i 137.47 160.75 121.91 138.74 151.67 128.52 138.68 117.46 134.63 137.06 74.19 42.54 137.53 139.65 137.41 150.56 32.13 39;23 137.47 160.75 138.74 151.67 52.81 138.68 117.46 65.80

465.14 450.46 424.65 412.83 426.34 452.84 465.14 450.46 424.65 412.83 426.34 452.84 465.14 450.46 424.65 412.83 426.34 452.84 465.14 450.46

Q(9), FH(B) and Fy(8) calculated in a similar manner are obtained as follows:

" Table 11.1 (d) •Unsteady Propeller Loading Calculations (Example 1)

8

Q(8)

FH(e)

Fy(e)

deg

kNm

kN

kN

344.02 328.46 322.30 331.48 347.48 . 353.85 344.02

3.91 16.90 22.39 15.10 1.68 -4.02 3.91

30.83 31.08 37.84 44.84 44.97 37.72 30.83

o 15 30 45 60 75 90

"-,------------~-------


319

Table 11.1 (d) (Contd.) ,'I

fz

8

Q(8)

FR(8)

Fv(8)

deg

kNm

leN

kN

105 120 135 150 165 180 195 210 225 240 255 270 285 300 315 330 345 360

16.90 328.46 322.30 22.39 331.48 15.10 1.68 347-48 353.85 -4.02 344.02 3.91 328.46 . 16.90 322.30 22.39 15.10 331.48 347.48 1.68 353.85 -4.02 344.02 3.91 328.46 16.90 22.39 322.30 331.48 15.10 1.68 347.48 353.85 -4.02 344.02 3.91

31.08 37.84 44.84 44.97 37.72 30.83 31.08 '37.84 44.84 44.97 37.72 30.83 31.08 37.84 44.84 44.97 37.72 30.83

As expected, all the four quantities, T(8), Q(8), FR(e) and Fv(e), can be repre sented by A + B cos(48 + c:) where c: is a phase angle, the other harmonics giving a zero net contribution when summed over all the four blades, Fig. 11.'2.

11.2

Vibration and Noise

The propeller in a ship is a source of vibration and noise. The unsteady forces due to the operation of the propeller in a non-uniform wake field give rise to the vibrations of the hull and the propeller shafting system. The propeller blades themselves may also vibrate. These vibrations may produce noise that is undesirable, particularly in warships. Propeller excited vibration occurs due to the pressure pulses of the pro peller being transmitted through the water to the hull. The propeller and ./ the hull also interact through the shafting sy~tem. The forces and moments

L


320 10.0

2.0

9.0

1.0

8.0 7.0 V ms-' a 6.0

\.

/

_. - -- I - - -" 90

180

270

360

-1.0

Vms-I

l

-2.0

~

~

Q"'"

' - -

--

- -

I'

...... ..... '

r -

90 180 degrees

1

--

-- I

270

360

o

e

50

.....T

I-

8 degrees'

-

" ~

o I...........

,/

a

f I ~

"

I'--..

-/

90

180

270

360

degrees

r--~..,.--.--.--r--,---r--,

40 1-t'=-\-I-ft-\-I--+t--\+-+t---\-1 30 20 FV kN 10 ~ kN 0 H _ 10 L-L.....::-':--'--'----'---=-=-::-'--=-:' o 90 180 270 360 degrees

e

Figure 11.2: Unsteady Propeller Loading (Example 1).

that act through the propeller shaft bearings are termed bearing forces, and include the effects of the weight of the propeller in water and the propeller inertia, the added mass, added inertia and damping due to the water around the propeller, and the propeller forces and moments. The propeller forces and ~oments, which can be divided into steady and cyclic components, are transmitted through the shaft bearings to the hull. The pressure pulses that are transmitted to the hull through the water are caused by the pressure field accompanying the propeller blades as they pass close to the hull and by cavitation. The pressure at a point due to the pressure field of a propeller in the absence of cavitation varies in a continuous manner with time. The varia tions in pressure are comparatively small, and depend on the thickness of the blades and their pressure distribution. The pressure pulse Po due to the propeller when there is no cavitation decreases sharply as the distance 2 5 5 from the propeller increases, Po being proportional to 5- . . The pressure pulses due to. cavitation depend upon the type of cavitation, the pressure fluctuations being the greatest for bubble cavitation, somewhat less for sheet cavitation and still less for tip vortex cavitation. The pressure fluctuations


321

depend upon 8 2v / 8£2 where v is the cavity volume and i the time, so that the pressure fluctuation produced by the sudden collapse (implosion) of a cavity is much larger than that produced by a growing cavity volume. The pressure change Pc due to cavitation varies inversely as the distance from the cavity, i.e. Pc is proportional to 8- 1 , so that the pressure pulse at a point due to a cavitating propeller is much stronger than that due to a non-cavitating propeller. The pressure at a point in the water is also influenced by the presence of a solid boundary: the pressure pulse at a point on a flat plate placed close to the propeller is nearly twice the pressure pulse at that point in the absence of the plate. This effect is represented by the "solid boundary factor" defined as the ratio of the pressure pulse at a point in the presence of the solid boundary to the pressure pulse without the solid boundary. The solid boundary factor depends upon the shape of the surface and for a ship hull is typically equal to about 1.8. In addition to the pressure fluctuations caused by the propeller, it is also necessary to take into account the pressure fluctuations caused by the vibration of the hull surface in water. The determination of hull pressure fluct.uations due to a propeller may be carried out by theoretical means using unsteady lifting surface theory. However, to obtain accurate results it is necessary to take into account wake scale effects, the solid boundary factor and the effect of hull vibration on the pressure. Hull pressures may, also be determined by model experi~ents in a cavitation tunnel. It can be shown by dimensional analysis that the pressure at a point in the water near the propeller is given by:

(11.11)

where 8 is the distance of the point from the propeller, and the other symbols have their usual meanings. In the experiment, all the parameters for the geometrically similar model are made equal to those for the ship except for the Reynolds number, and then: PS PM

=

PS A PM

(11.12)

the suffLxes Sand M referring to the ship and the model, A being the model scale.

322


Hull surface pressures caused by the propeller may also be predicted by empirical methods based on an analysis of a number of ships. In one such method, due to Holden, Fagerjord and Frostad (1980), the hull surface pres sures at a point are given by: 2 2 Po = 12.45pn D Z-1.53

Pc

c) (25)

0.098 p n 2 D 2 V ( W max

P = (P5

~7

1.33

- W )

D

b 0"-Q.5

-a

(25) D

-al

(11.13)

+ P; )0.5

where: P

-

pressure at a point due to the propeller;

Po

-

pressure due to the propeller without cavitation;

Pc

=

pressure due to cavitation;

p

density of water;

D

= = =

Z

-

number of blades;

to.7

-

blade thickness at 0.7R;

5

= distance of the point from 0.9R when the blade is upright;

V

=

n \

propeller revolution rate;

propeller diameter;

ship speed;

maximum value of the wake fraction in the propeller disc;

lL'ma..x

mean effective wake fraction for the ship;

W

--

0"

= cavitation number based on the tangential velocity at 0.7 R.

r

323

Some .Miscellaneous Topics

t

r,

a

=

s s 1.8 + 0.8 D for D :S 1.0

s 2.6 for D ~ 1.0

I·

al

=

1.7-1.4

s

D

s = 1.0 for D

s for D :S 0.5

~

0.5

(pitch x camber) at 0.95R (pitch x camber) at 0.80R

b

These formulas have been criticised for not having a proper theoretical basis. In. any case, such empirical formulas give only approximate values and large differences can occur between the values predicted by such formulas and those obtained in practice. In addition to exciting hull vibration, the propeller blades may themselves undergo flexural and torsional vibration. The higher modes of propeller blade vibration after the fundamental mode and the first flexural and tor sional modes are extremely complex due to the shape of the blades: non symmetrical blade outline, variable blade thickness along the radius and along the chord at each radius, and the twist of the blades due to the vary ing pitch angles along the radius. The surrounding water also influences propeller blade vibration significantly: the mode shapes and frequencies of a propeller blade vibrating in air are quite different from those of a blade vibrating in water. Simple empirical formulas proposed by Baker (1940-41) are often used for estimating the frequencies of the fundamental flexural and torsional blade vibration:

iJ

0.305 to [ ~ t m g E ] 0.5 ( R - ro ) 2 ~n to Pm (11.14)

it

0.92 to.s Co

------

R - ro CO.s em

where:

!J

it

fundamental frequency of flexural vibration in air;

=

fundamental frequency of torsional vibration in air;

L-------


324

R

;:::

propeller radius;

ro

;:::

radius of root section;

to

;:::

thickness of root section;

tm

;:::

mean of the blade section thicknesses; blade thickness at a radius of O.5(R + ro);

to.s CO Ctn

;:::

chord length ofth~ root section;

=

mean chord length of the blade;

CO.5 =

chord length at a radi~s of O.5(R + To); acceleration due to gravity;

Pm

= =

E

=

modulus of elasticity of the propeller material;

G

=

modulus of rigidity of the propeller material.

g

density of the .propeller material;

The blade frequencies in water are about 65 percent of the correspond

ing frequencies in air. The material of the propeller affects blade vibration

through its damping properties. Commonly used propeller materials have

low.damping, but special materials having high damping characteristics are

available. Propeller blade vibration may be studied using the finite element

technique, but it is necessary to model the conditions at the blade root

properly and to take into account the effects of added mass and inertia. In

propellers running at high rpm it is also necessary to consider the effects of

centrifugal force.

The noise generated by the hull, the machinery and the propeller is of

great importance in a warship. The noise not only betrays the position of

the ship and helps in its identification but also interferes with the ship's own

sensors. Noise is also important in oceanographic research vessels where it

may interfere with the operation of sensitive ini3truments. The importance

of minimising noise is growing in merchant ;hips also since noise is' a health

hazard. An important source of noise in a ship is the propeller.

I I 11<

~. ~

~,

~ ~

t ~ ~

~


325

The speed of sound in water is more than four times the speed of sound in air, and therefore the wavelength of a sound wave of a given frequency is greater in water than in air. The transmission of sound in water depends upon the frequency: high frequency sounds are attenuated more rapidly than low frequency sounds, i.e. low frequency sounds travel farther. The noise level is measured in decibels (dB) by the ratio of the power P of the sound source to the power Po of a reference source, or in terms of the ratio of the sound pressure P to a reference pressure Po:

L p = 10 log 10

(~)

(11.15)

The reference pressure in water is taken as 1 J.L Pa A10- 6 N per m 2 ). Acoustic measurements are usually given for octave or one~third octave bands, i.e. the ratio of the highest frequency to the lowest frequency considered is 2 or 2 1 / 3 respectively. " Propeller noise arises basically from four causes: _ _ - The~cceleration imparted to the water'by the propeller, - The rotating pressure field of the propeller, - The periodic fluctuation" in the size. and shape of cavities caused by the propeller operating in a non-uniform flow, and - The collapse of cavities.. The noise due to the first two causes is generally small and of the same level as the noise due to the machinery or that due to the passage of the hull through the water (boundary layer noise). However, once cavitation starts, propeller noise becomes predominant; hence the importance of delay ing cavitation as much as possible. The noise due to the action of a propeller without cavitation consists of sound at the blade passing frequency (number of blades x propeller revolution rate) and its multiples, Le. there are sev eral distinct frequencies, usually below the audible range (20 Hz). There is also a broad band noise (comprising sounds with a wide frequency range) at higher frequencies due to turbulence and vortex shedding. When cavita tion occurs, the collapse of bubble cavities produces noise covering a wide


326

range of frequencies, up to 1 MHz. The greatest contributions to noise are made by sheet cavitation, cloud cavitation and tip vortex cavitation. The distribution of noise 'levels for different frequencies is given in the form of a noise spectrum Lp(f), which gives the sound pressure level as a function of frequency. Sounds of distinct frequencies appear as vertical spikes in a plot of the noise spectrum whereas broad band noise appears as a continuous line, Fig. 11.3. The noise spectrum of'a ship helps to identify it and is therefore sometimes called its signature. 140 r - - - - - - - - - - - - - - - - - - - - - , /--

120

I

/

....

"

"

100

"

CAVITAllNG PROPELLER

"" "

NON-CAVITAllNG I"ROPEu.ER

SOURCE SPECTRUM

""

"

' ........ ,

60

LEVEL

(dB ref. 11lPo) 60

1--

10

---1._-'--_ _-1...

100 FREQUENCY, Hz

1000

..1.

10000

--'

100000

Figure 11.3: Propeller Noise Spectra.

Tlw prediction of propeller noise may be done with the help of empirical formulas, such as one due to Ross (1976): (11.16)

where the reference pressure is 1 f-L Pa, the band width is 1 Hz, and the sound pressure is measured at a distance of 1 m from the source of sound. The Ol'etical methods are as yet incapable of predicting cavitation noise, and predictions of propeller noise are usually made with the help of experiments in a cavitation tunnel. The reverberation of noise in a cavitation tunnel is a major problem. There are also problems of scaling up the noise measur"ed on the model scale to full scale.

Some :Miscellaneous Topics

327

Olrection of

Revolution SECTION A-A

~

0"2m

~

o

0.0050 + 0.001

b

0.10

Figure 11.4: Anti-singing Edge in a Propeller Blade.

A particular type of noise occurs when the propeller blades vibrate in the audible frequency range. This is known as the singing of propellers. Singing is probably due to the blades beil1g set into vibration by boundary layer separation near the trailing edges, resulting in the shedding of vortices alternately from the face and the back of the blades. The phenomenon of singing is unpredictable since it is often found that of two propellers of the same design one sings while the other is silent. The singing of propellers takes many forms, and may occur at different frequencies from low to high in a particular range of propeller rpm. Propeller singing is usually unimportant except when it occurs in the normal range of operation of the ship. If necessary, the propeller blades can be given a special "anti-singing" edge in which the trailing edge in the outer part of the blade has a sharp wedge shape, Fig. 11.4. This eliminates the alternate shedding of vortices and the resulting blade vibration because the boundary layer separation occurs at fi.xed points near the trailing edge.

11.3

Propulsion in a Seaway

The power required to propel a ship at a given speed increases with the severity of the conditions in the sea. At a given power, the speed attained

L----

328


decreases as the sea state increases. This decrease in speed at constant power due to the sea conditions is termed "involuntary speed loss". If the severity of the sea conditions become excessive, it is necessary to reduce the power to limit the motions of the ship and prevent damage. The decrease in speed due to a deliberate reduction in power is termed "voluntary speed loss". There are two approaches to estimating the effect of sea conditions on ship propulsion: (ij self-propulsion tests with models in waves, and (ii) cal culations based on measurement or calculation of the added resistance of the ship in waves and estimated values of the propulsion factors. In carry ing out self-propulsion tests in waves, it is necessary to take into account the response of the propulsion plant to the fluctuations in the loading of the propeller in wayes. Diesel engines have constant torque characteristics, so that an increase in the loading causes a decrease in the propeller rpm and delivered power at constant torque. Thrbines, on the other .hand, have constant power characteristics so that increased propeller loading causes an increase in the torque and a decrease in the propeller rpm at constant de livered power. These characteristics must be incorporated into the control system of the drive motor used in the model self-propulsion tests in Waves. In calculating the effect of the seaway on the propulsion of a ship, the added resistance in waves and the wind resistance are calculated and added to the calm water resistance. It is usually assumed that the propulsion factors (wake fraction, thrust deduction fraction, and relative rotative efficiency) are independent of the wave frequency and have the same values as in calm water. at a propeller loading equal to the average loading in waves. It is also assumed that the propeller characteristics (KT, KQ, J) in waves are the same as. in calm water. However, these assumptions are obviously untenable if the propeller emerges out of water due to ship motions and if there is air drawing. Self-propulsion experiments and the calculation of power in waves are car ried out for regular waves for a range of wave frequencies. An "added power .operator" oP/ (2 is determined as a function of the wave encounter frequency WE, oP being the increase in average power in waves over that in calm water and ( the wave amplitude. The mean added power in irregular long crested seas is then given by:

It


329

where S((WE) is the encounter spectrum of the seaway and represents the distribution of wave energy as a function of the frequency of wave encounter. In this way, self-propulsion tests or added power calculations for different ship speeds and headings (relative to the waves) can be used to predict power as a function of speed for various sea conditions represented by their energy spectra. Example 2 , The effective power of a ship in calm water js as follows:


14.0 2100

14.5 2375

15.0 2674

15.5 2999

16.0 3352

16.5 3733

The ship has a propulsion plant giving a delive~ed power of 4900 kW at 120 rpm to the propeller, which has a diameter of 5.5 m and a pitch ratio of 0.8. Determine the speed of the ship in calm water. What will b~the involuntary speed loss in waves if the ayerage increase in effective power over that in calm water is 40 percent and (a) if the propulsion plant of the ship consists of a diesel engine directly connected to the propeller, and (b) if the propulsion pl~nt consists of a geared steam turbine? Calculate the propeller rpm, torque and delivered power in the two cases. Assume that the propeller open water characteristics and the propulsion factors are not affected by the waves, and are as follows:

KT

=

0.3415 - 0.2588 J - 0.1657 J2

10Kq

= 0.4021- 0.2330 J -

0.2100 J2

Wake fraction = 0.220, thrust deduction fraction = 0.190, relative rotative effi ciency = 1.030, thrust identity.

PD = 4900k\V

nl

= 120 rp~ = 2.000 s-1 (rated rpm)

R/(I-t) pD2 (1- w) y2

=

=

1

1.025

X

D = 5.5m

5.5 2 x (1 - 0.190) x (1 - 0.220)2

PE y3

=

1

PE

15.2800

y3

From this, the values of J can be determined for- different speeds Y using the open water characteristics: 2

0.3415 (] )

-

0.2588] - (0:1657 +

I~;)

= 0


330

and the values of 10KQ calculated. The speed of the ship will be given by that value of V at which the KQ from the open water characteristics is equal to the KQ calculated from the delivered power: (a) For the diesel plant: Q

=

PD

==

270 nl

4900 = 389.930 leN m 271" x 2.00 .

and this is constant. so that: 389.930 'x 1.030 X n 2 x 5.5 5

QTJR

K Q == pn 2 D5 == 1.025

n ==

VA

JD ==

=

0.077855 n2

(1- 0.220) V V (1-w)V = 0.141818 J == J x 5.5 JD

PD == 271"nQ

(b) For the geared turbine plant: PD is constant and:

K' _ Q -

PD1}R == _ _4_9_00_X_l_.0-::-3_0_= 27. P n 3 D5 271" x 1.025 x n 3 X 5.5 5

=

0.155710

n3

Table 11.2 (a) ~

Calm Water Propulsion Characteristics (Example 2)

\

Vk

14.0

14.5

15.0

15.5

16.0

16.5

7.2016

7.4588

7.7160

7.9732

8.2304

8.4876

PEkW:

2100

2375

2674

2999

3352

3733

J{T/P:

0.3680

0.3746

0.3809

0.3872

0.3935

0.3996

J

0.5934

0.5908

0.5884

0.5860

0.5836

0.5814

0.1296

0.1308

0.1319

0.1330

0.1340

0.1350

lOKQ

0.1899

0.1911

0.1923

0.1935

0.1946

0.1957

ns- 1

1.7211

1.7904

1.8599

1.9297

2.0000

2.0705

rpm

103.3

107,4

111.6

115.8

120.0

124.2

10K'Q 10K'Q

0.2628

0.2429

0.2251

0.2091

0.1946

0.1816

0.3054

0.2713

0.2420

0.2167

0.1946

0.1794

ms- 1 :

KT

(a) (b)

:


331 Table 11.2 (b)

Propulsion Characteristics in Waves (Example 2)

40 percent· increase in Effective Power

14.0

14.5

15.0

14.176

14.365

7.2016

7.4588

7.7160

7.2923

7.3894

2940

3325

3744

3073

3218

0.5152

0.5244

0.5333

0.5185

0.5220

J

0.5432

0.5405

0.5379

0.5422

0.5412

KT

0.1520

0.1532

0.1543

0.1525

0.1529

10KQ ns- I

0.2136

0.2148

0.2160

0.2140

0.2145

1.8801

1.9570

2.0342

1.9073

1.9363

rpm

112.8

117.4

122.1

114.4

116.2

(a)

10K'Q

0.2203

0.2033

0.1882

0.2140

(b)

10K'Q

0.2343

0.2077

0;1850

Vk

ms-I: PEkW:

KT

fJ2;

QkNm:

389.93

402.75

PDkW:

4673

4900

Diesel

Turbine

"Diesel

Turbine

Involuntary speed loss, k

1.824

1.635

Propeller rpm

114.4

116.2

Torque, kNm

389.93

402.75

4673

4900

Delivered. Power, kW

11.4

0.2145

Propeller Roughness

When a ship goes into service, its hull and propeller surfaces get progressively rougher. The increase in hull surface roughness causes an increase in the effective power of the ship. The roughness of the blade surfaces of a propeller affects its performance characteristics. There is an increase in the torque


332

coefficient KQ and a decrease in the thrust coefficient KT, which together cause a significant decrease in propeller efficiency. The effect of propeller roughness depends on both the amplitude of the roughness and its texture. A measure of propeller surface roughness IS Musker's h' parameter which is expressed in many ways, one of which is:

h' = 0.0147 R~2.5Pc

(11.18)

where R a 2.5 is the root mean square roughness height in microns over a 2.5 mm length and Pc is the peak count per mm as determined by a portable roughness gauge. A simpler approach to deterinin'e propeller blade surface roughness is to compare the propeller surface with the Rubert compara tor gauge, which has six representative surfaces of different roughnesses as follows: "A

Surface

h' microns:

1.1

B

.c

D

E

F

5.4

17.3

61

133

311

If the roughness varies over the propeller surface, an average propeller rough ness may be calculat':ld by dividing the surface into a number of parts, de termining the roughness parameter for each part and finding a weighted average taking into account the area of each part and the radius at which it is located. Thus, if the propeller blade surface is divided into radial strips,

the aveFage propeller roughness is given by:

:I

'I,

-

I

kp =

I cr 3 h' dr I cr3 dr

-1

i

(11.19)

where c and r are the mean chord and mean radius of each strip and h' the roughness parameter for that strip. The drag coefficients of a propeller blade section with smooth and rough

surface are given according to Mishkevich (1995) by:

CDs

= 0.05808 ( 1 + 2.3~)

R;;~·1458

I

J


333

(11.21)

where t and c are the thickness and chord of the representative blade sec tion of the propeller, i.e. at O.7R or O.75R, and Rnc is the Reynolds number based on the resultant velocity and chord of the section. The change in the drag coefficient tlCD for differing degrees of roughness can then be deter mined. One may alternatively use the formulas in Eqn. (8.11), Chapter 8, to determine tlCD . . The roughness of the propeller blade surface also causes a change in the lift coefficient of the blade section. According to Mishkevich, this change tlCL is given by: (11.22) or more simply according to Townsin, Spencer, Mosaad and Patience (1985) by: (11.23) These changes in the lift and drag coefficients of the propeller blade sec tions cause changes in the thrust and torque coefficients which, it can be shown, are given by: 11"2 11.0 _x c 2 cos (/h - (3 ) - -z (3 [tlCL cosf3[ 4 D cos 2

2

tlCD sin{3[] dx

Xb

11.0 _x3 . cos ({3[-{3) -z f3 [tlCL sin{3[ + tlCD cos{3[] dx 8 D cos 7[2

C

2

2

Xb

where:

z

=

number of blades; non-dimensional boss radius;

Xb

c

=

chord at radius rj

D

=

propeller diameter;

x

1-----

non-dimensional radius r / R;

(11.24)


334 {31

-

hydrodynamic pitch angle including induced velocities;

{3

-

hydrodynamic pitch angle excluding induced velocities. (See Fig. 3.4)

The effects of propeller roughness on the performance characteristics can thus be calculated. These expressions can be greatly simplified by making various assumptions and approximations. If it is as'sumed that the blade section at x = Xl is representative Of the whole propeller and that: dKT dx (11.25)

L

):

then:

where m(xl) is a coefficient that depends on the value of which is then given by: (1 -

Xb

)O.5( 8 + 12 Xb + 15 x~ (1- xdO. 5

xt )

One may write:.

where c,{3,{3I,b.CL and b.CD all refer to the section at

Xl.

Xl

chosen, and

(11.26)

Some bJiscellaneous Topics

335

Still further simplifications are possible if one makes the following assumptions:

J = 0.8

tan 131 =

P~l)

(11.28)

P(XI)/ D

(11.29)

1r Xl

which imply that the propeller works at a 20 percent nominal slip ratio and zero angle of attack. Since: tanl3

=

J

(11.30)

the values of 131 and 13 can be calculated for a given propeller, and numerical values of f)..KT and f)..KQ determined for a given average propeller roughness kp . However, it is found that cos 2 (131 ~ f3)cos 131/ cos2 13 varies only slightly with pitch ratio, having values of 1.003 at P/ D = 0.4 and 1.061 at P/ D = 1.6 based on Eqns. (11.28), (11.29) and (11.30), and an average value of 1.026 over this range of pitch ratio. Then:

(11.31)

If one takes the representative section of the propeller to be at and the root section at Xb = 0.20, m(xI) = 0.6663 and:

Xl

= 0.75

(11.32)

ti.KQ = (0.1510 ti.CL

~ + 0.3558 ti.'cD)

; Z


336 Example 3

A four-bladed propeller of 5.0 m diameter and 0.8 pitch ratio has a chord of 1.375 m and a thickness of 0.0675 m at 0.75R. The propeller runs at 150 rpm. The thrust and torque coefficients of the propeller when new are:

J :::: 0.640

KT

::::

0.1080

10KQ = 0.1670

Assuming that the new propeller has a smooth surface, determine the thrust and torque coefficients when the propeller roughness is 300 microns.

Z c

D KTs

p

D :::: 5.0m

::::

4

=

1.375 5.00

::::

D

t

0.275

= 0.8 ::::

C

= 0.1080

10KQ$

C ::::

0.0675 1.375

::::

1.375m

t

::::

0.0675m

0.0491

n

::::

150 rpm = 2.5 s-l

= 0.1670

(Subscripts sand r for smooth and rough respectively)

1Jos

=

KTs J K Q $ 27f

=

0.1080 0.640 . x - - :::: 0.6857 0.01670 27f

VA :::: JnD = 0.640 x 2.5 x 5.0 :::: 8.000ms- 1 V~ :::: V~

+ (0.75 7f n D)2

= 8.000 2 + (0.757f x 2.5 x 5.0)2

:::: 931.446 m 2 S-2

, V R = 30.520ms- 1 :::: 30.520 x 1.375 :::: 35.323 1.188 X 1O~6 CDs ::::

0.05808 (1+ 2.3

X

106

~) R;;~·1458

:::: 0.05808 (1 + 2.3 x 0.0491) x (3.5323 x 107 )-0.1458

:::: 0.005128


337 1.375

== (1 + 2.3 x 0.0491) x ( 2.34 + 1.121 log 300 x 10

)

-2.5

6

== 0.010557

~CD

== CDr - CDs == 0.005429

6.C L

== -1.28826.d}j94 == -0.009564

6.KT

== (0.94886.C L

-

~) ;

0.4027 6.CD

Z

== [0.9488 x (-0.009564) - 0.4027 ~ 0.005429 x 0.8] x 0.275

X

4

X

4

i

== -0.0119 6.KQ

==

(0~1510 6.CL ~ + 0.3558 6.Cf ) ;

== [0.1510 X (-0.009564)

Z

"

X

0.8:+ 0.3558

X

0.005429 J X 0.275

== 0.00085 KTr.== K Ts K Qr

+ 6.KT ==

0.1080 - 0.0119 == 0.0961

== K Qs + 6.KQ = 0.01670 + 0.00085 == 0.01755

TJOr == KTr.:!..- == 0.0961 KQr 211" 0.01755

X

0.640 211"

== 0.5578

As this example shows, propeller roughness can have a significant effect on propeller efficiency, and it is necessary to service the propeller properly every time the ship is dry-docked. Proper servicing of the propeller includes: - rectification of any damage - correction of edge deformation - light grinding and polishing of the blade surfaces to produce a finish as good as new, i.e. less than 3 microns roughness. Great care should be taken near the leading edges and nea!' the blade tips to ensure that the blade shape is not altered during grinding.

=l---


338

11.5

Propeller Manufacture

The manufacture of marine propellers is a specialised activity requiring the use of facilities for making large castings and then machining them to the specified degree of accuracy. It is necessary to ensure that the castings are free of defects and the finished propeller is in accordance with the shape and dimensions specified by the propeller designer. An inaccurately man ufactured propeller may have a low efficiency and suffer from cavitation, vibration, ero5ion and noise. The mould for casting a small propeller (less than 2.5 m diameter) may be made using a wooden pattern of one blade, with the mould being in two parts. For large propellers, the mould for the face of the propeller blades is made by "sweep moulding". A sweeping board or "knife" is used to form a helicoidal surface by rotating the knife about an axis while guiding it along a template in the form ofa helix of the appropriate pitch. This produces a helicoidal surface of constant pitch. If the inner and outer ends of the knife are constrained to follow different pitch templates when the knife is rotated about the a.'65. a helicoidal surface with a linearly varying pitch is obtained. By displacing the knife axis from the propeller axis, it is possible to obtain a hyperbolic pitch variation along the radius. The mould itself is made of sand and cement with appropriate internal steel reinforcements. Once the mould for the faces of the propeller blades is made, templates for the radial sections are set up at a number of radii. The helicoidal surface for each blade face is also filled with a sand-cement mixture to incorporate the required pitch variation and the washaway on face. The radial section templates are then used to make a sand blade shape on each helicoidal surface of the face mould. The back mould is cast in sand and cement on the face mould with its sand bl_ades. After the cement moulds have cured, the two parts are separated and--{air~d (i.e. the surfaces are made smooth and blemishes removed). Runners ana risers for the casting are provided. Trailing edge recesses are also made if necessary to ensure that the liquid metal flows upward during the casting process and no air is trapped inside. The two parts of the mould are then put back together and firmly held by girders and rings. The metal of which the propeller is to be made is melted in a furnace and the molten metal is then poured into the mould through the runners provided


339

at the sides. It is usually necessary in large propeller castings to pour the molten metal in two or three stages to counteract the effect of contraction while cooling. The molten metal is at a sufficiently high temperature to ensure that there is complete fusion (950-1100° C for Manganese Bronze and 1050-1200° C for Aluminium Bronze). ' After the propeller casting has cooled, the mould is broken and the casting taken out. The casting is checked for its soundness by dye penetrant or ultrasonic tests, and the mechanical properties and chemical composition of the material are determined from specimens taken from the runners and risers. The next step is to determine the position of the propeller axis and a ref erence plane normal to the axis such that the finished blades lie within the casting. Once this has been done, the propeller bore is machined in a horizon tal boring machine using plug gauges supplied by the shafting manufacturer. After the propeller bore has been machined, the blades are machined to their specified shape. At one time, thil? was done by special planing machines for finishing the propeller blade face. This has ,been superseded by a manual method which is found to be more economical. A large number of points on the face and the back are marked, and at each point the amount of material to be removed is determined. The unwanted material is then removed by pneumatic chisels, and the blade surfaces finished by grinding using various grades of grinding wheels. I

In recent years, propeller manufacturers have been switching over to the use of numerically controlled milling machines for finishing propellers. CNC (computer numerical control) five-axis machines with as-machined ar;ima tion and adaptive control are being increasingly used. Animation and solid modellers provide a computer simulation of the cutter paths and display the material to be removed and the final finished product. With adaptive con trol, the machining parameters are sensed and automatic adjustments made to cutter speeds and feeds. The use of computer aided design and man ufacture (CAD/CAM) in manufacturing propellers has many advantages: greater accuracy and savings in time, labour, space and cost. After the propeller has been machined, it is checked for dimensional accu racy: diameter, pitch and blade thicknesses at various radii, shapes of the root fillets and the leading and trailing edges, the axial alignment of the blades and their angular spacing, and the su'tface roughness. Finally, the


340

I--- AXIS

WAX CYLINDER

-_.r

MOULDING KNIFE ./

HELICOIDAL SURFACE

Figure 11.5: Moulding a Helicoidal Surface. \

prop,~ller is statically balanced. The manufacturing tolerances for marine

propellers specified by the International Standards Organisation (ISO) are given in Table 11.3. Model propellers for use in model experiments are made in a manner sim ilar to ship propellers. A wax cylinder is shaped into helicoidal surfaces, one for each blade, using a knife revolving about an axis while moving along a helical template of the appropriate pitch, Fig. 11.5. A plaster of paris mould is made from this wax surface for the faces of the propeller blades. Tem plates representing the radial sections of the model propeller are set up at a number of radii, and clay blades made on each helicoidal surface. Another plaster of paris mould is cast over the clay blades to form the mould for the

--~-'""""'":7-' --"'-'~;::;~:r~

Table 11.3

~

Summary of Manufacturing Tolerances for Ship Propellers according to ISO No

ITEM

CLASS S

I

II

III

Very High

High

Medium

Wide Tolerances

1.

."vfanuJacturing Accuracy

2.

Radius Tolerance as percentage of propeller radius with a minimum of (mm)

±0.2 1.5

±0.3 1.5

±OA

Pitch a) Local Pitch

Tolerance as percentage of design pitch

at each radius Minimum (mm)

±1.5 10

±2.0 15

±3.0

20

- - ±1.0

±1.5

±2.0

±5.0

7.5

10

15

25

±0.75

±1.0

±1.5

±4

7.5

10.0

20.0

±0.75

±1.0

±3.0

7.5

15.0

E

~

fi

g
0

3.

, b) Mean pitch at each radius of each blade

Tolerance as percentage of design pitch at each radius

Minimum (mm) c) Mean pitch of each blade

Tolerance as percentage of design mean pitch

Minimum (mm) d) Mean pitch for all the blades

Tolerance as percentage of design mean pitch

Minimum (mm)

&i

5.0 ±0.5 4.0

5.0

2.0

±0.5 2.5

....~

'15

&l

w

....I!>

~

Table 11.3 {Contd.) Summary of Manufacturing Tolerances for Ship Propellers according to ISO ITEM' '

No

CLASS S

4.

tl: t"

Blade thickness a) PIlls tolerance as pcrcentap;1' of the' local

I

II

III

+2.0

+2.:;

+1.0

+(i.O

0.5

1.0

2.0

3.0

-1.0

-1.5

-2.0

-4.0

-0.5

-l.0

-1.5

-2.0

Blade width Tolerance as percentage of D I Z Minimum (mm)

±1.5 4

±2.0 7

±3.0

±5.0 12

Angular spacing of blades Tolerance in degrees

±1

±1

±2

±2

Axial position of bla.dcs Tolerance as percentage of diameter D

±0.5

±l.O

±1.5

±1.5

I.!JicklJes:-;

Minimum (mm) b) Minus tolerance as percentage of the local thickness Minimum (mm) 5

6. 7.

8.

Surface finish Maximum roughness in microns

3

6

10

12

25

Measurements should be made at: - a section near the boss and at r I R = 0.4,0.5, 0.6, 0.7, 0.8, 0.9 and 0.95 for Class S and Class I propellers - a section near the boss and at r I R = 0.5, 0.7 and 0.9 for Class II and Class III propellers.

tl:l ~

o·

~ 'S'

1 c Si

0'

l:l

'It ~~--

~;q_

'V".

__~':

'0.

--..-.• ~~.....

i

.~:...

;;-',.,,:.::.;.._,__ '.'

--'-"·;'-'-<--'K?'''#-·1'-\';'l·I-::-~~;,!?~-~~~?f-.f~.~-=fl~T'.:;:tf;,}?~2:~$'~"·IilII

-------"-

Some 1vliscellaneous Topics

343

back surfaces of the propeller blades. After this has set, the two parts of the plaster mould are separated and faired. Runners and risers for casting the metal are made, and the two parts of the mould fitted firmly together. The molten metal, usually white metal or aluminium alloy, is then poured into the mould and the model propeller casting obtained. . The finishing of the model propeller casting is $:arried out in several ways. The commonest method is to use a special propeller point drilling machine in which a large number of points are drilled on the face and the back of each blade to their specified cylindrical polar coordinates (r, (J, z). The blades are .then finished by hand grinding and filing. Another method is to use a copy milling machine with an accurately made blade pattern. However, today the trend is to use CNC five-axis milling machines even for model propellers, but this usually requires the blades to be suitably supported to prevent their deflection under the load imposed by the milling cutter. Instead of machining a casting of the model propeller, one may start with a solid block of metal and machine it to the final,.model propeller shape. Typical manufacturing tolerances for model prope\lers are given in Table 11.4. TablellA Manufacturing Tolerances for Model Propellers

11.6

Dimeter

±O.75 mm

Mean pitch at each radius:

±O.5 % of design value

Blade thickness

±O.125 mm

Blade width

±O.2 mm

Acceleration and Deceleration

The acceleration and deceleration of a ship using its propulsion system involve rather complex phenomena and are difficult to calculate. The prin ciples involved are, however, quite simple. Consider a ship undergoing acceleration. The net force causing the acceleration is: F

=

(1 - t ) T - RT

=

,

dV

'(1 + c) t:.. dt

(11.33)


344

where: t

;:;::

thrust deduction fraction;

T

;:;::

propeller thrust (positive forward);

RT

:::::

total resistance of the ship;

c

- added mass coefficient;

J:).

- displacement (mass) of the ship;

V

=

ship speed;

t

-

time.

\

Il

The resistance of the ship is a ftinction of the ship speed while the thrust depends upon the speed of the ship, the propeller revolution rate, the propul sion factors and the open water characteristics of the propeller. The added mass coefficient depends upon the hull form of the ship. The time required for the speed of the ship to change from V1 to V2 is:

(11.34) and the distance travelled by the ship in this period is: \

(11.35) These equations may be used to calculate the acceleration, stopping and reversing characteristics of a ship. Hbwever, the determination of the vari ous quantities involved in these equations requires propulsion and open water tests covering the four quadrant operation of the propeller, i.e. both direc tions of revolution and speeds of advance, over the required speed range, as well as resistance tests for both directions, forward and aft. It is also neces sary to know the machinery characteristics involved in limiting the propeller revolution rate at different ship speeds and in stopping the revolution of the propeller in one direction and starting it in the opposite direction. Finally, it must be noted that when a ship is being stopped by reversing its propeller

Some 1Yfiscellaneous Topics

345

it tends to follow a curved path, and this is not taken into account in the foregoing equations. Example 4

A ship of 10000 tonnes displacement has an effective power as follows:

Speed, knots Effective Power, kW:

4.0 27.2

12.0' 16.0 1420.0 4000.0

8.0 329.9

20.0

8931.7

The propeller has a diameter of 6.0 m and a pitch ratio of 0.7, and, its open water characteristics are given by:

KT

:=

10KQ =f 0.3187 - 0.2114 J - 0.2004 J2

0.2974 - 0.2536 J - 0.1840 J2

The propulsion factors, which may be assumed not to vary with speed, are as follows: wake fraction 0.250, thrust deduction fraction 0.200, relative rotative efficiency' = 1.050, based on thrust identity. The ship is propelled by a diesel engine of maximum rating 6000 kW at 120 rpm directly coupled to the propeller, the shafting efficiency being 0.980. Calculate the accelerating force on the ship at different speeds assuming that the maximum rated torque of the engine is not to be exceeded. If the added mass coefficient of the ship is 0.050, determine approximately the time taken for the ship starting from rest to reach a speed of 16 knots, and the distance travelled before this speed is attained.

=

=

A

=

10000 tonnes

KT

=

0.2974- 0.2536 J - 0.1840 J2

w = 0.250

D

c :::: 0.050

t k:

0.200

\

=

'l/R

=

=

1.050

=

2.0s- 1

=

6000kW

PD

::::

PB'l/S :::: GOOO x 0.980 :::: 5880kW

Q

lOKQ

PD = 27rn ::::

::::

10Q'l/R pn 2 D5

120rpm

Thrust Identity

'l/s

5880 :::: 467.916kNm 211" X 2.0 ::::

10

X

467.916

1.050 n 2

X

1.050 6.0 5

X

P

. D :::: 0.7

10KQ :::: 0.3187 - 0.2114 J - 0.2004 J2

PB

n

6.0m

::::

0.61642 n 2

=

0.980


346

v

= 0

"A =

PE = 0

0

0.61~44 = 0.3187

10 KQ =

=0

RT

J

=0

n = 1.3907 s-1 = 83.44 rpm

n

T = KTpn 2 D 4 = 0.2974 x 1.025 x 1.39072 x 6.0 4 = 764.1kN F

(ii)

= (1 -

t) T - RT

= (1 -

0.200) 764.1 - 0 = 611.3kN

v = 4.0k = 2.0576ms- 1 PE = 27.2kW v = (l-w)V = (1 -0.025) x 2.0576 = 1.5432ms-1

RT =

J =

10J~Q

PE V

27.2

2.0576

= - - = 13.219kN

VA nD

=

= 0.3187

1.5432 n x 6.0

(~ ) 2 _

\

= 5.8068

n =

n

~ _ 0.2004

0.2572 2 = 9.3183

n2

1

0.2114

. :

n 2

0.61642

J

=

.,

0.2572

0.2572 J

J. = 0.1722

=

K T = 0.2483

0.2572

1 0.1722 = 1.49355-

= 89.61rpm

T '= J(rpn 2 D 4 = 0.2483 x 1.025 x 1.4935 2 F

= (1 -

10 KQ = 0.2764

X

6.0 4 = 735.6kN

t) T - RT = (1 - 0.200) 735.6 - 13.2 = 575.3kN

Similarly, for the remaining speeds one obtains:

V knots: nrpm FkN

8.0 97.80 481.8

12.0 107.80 300.2

The integrations are performed as follows:

16.0 119.36 6.2

20.0 132.18 -638.2


~"

V

1000 P

F

1000 V -p-

8M

f(i)

/(5)

ms- 1

kN

0

0.0000

611.3

1.6359

0

1

1.6359

0

4.0

2.0576

575.3

1.7382

3.5766

4

6.9528

14.3064

8.0

4.1152

481.8

2.0756

8.5413

2

4.1512

17.0826

12.0

6.1728

300.2

3.3311

20.5623

4

13.3244

82.2492

16.0

8.2304

6.2

162.1271

1334.3709

1

162.1271

1334.3709

188.1914

1448.0091

k

,.

347

i,= A(l+c)

l

v, 1 F dV

VI

= 10000 ( 1 + 0.050) x

1

'3

188.1914 __ x 2.0576 x.. 1000 tonnes ms 1 kN 1

= 1355.3s

= 10000 ( 1 + 0.050) x

1

'3

x 2.0576 x

1448.0091 _ 1000 tonnes m 2 S-2 kN 1

= 10428m

The integrations using Simpson's Rule are not very accurate because as the steady speed is approached the acceleration reduces asymptotically to zero and the inte grands tend to infinity.

11. 7

..

J...---

Engine-Propeller Matching

In the design of the propulsion system of a ship, it is importan~ to match the ship, the propeller and the propulsion machinery so that the propui~i~n sy~'t~m: as a wh~le operates in the optimu~ manner. The task of opHriiising


348

the propulsion system is difficult because the power-speed characte.ristics of the ship and propeller taken together are very different from those of the engine. The power-speed characteristics of the ship and propeller moreover change with the loading of the ship and the sea conditions, and with time as the hull and the propeller get progressively rougher due to fouling, corrosion . and possibly cavitation erosion. In selecting the propulsion plant for a ship and designing its propeller or propellers, it 'is necessary to ensure that the /' desired ship speed is achieved without overloading the engine or exceeding its rated rpm in the varying operating conditions of the ship. If the engine and the propeller are not properly matched, the life of the engine may be reduced, maintenance costs may be higher and the fuel consumption may be greater. The relation between the power and the speed (rpm) for different types

of engines is illustrated in Fig.n.6. :While the maximum power output of

a steam turbine remains more or less constant as its speed is changed, the

maximum power of a d.c. motor decreases slightly as speed is reduced. The

maximum power available from a diesel engine is approximately proportional

to its rpm, the diesel engine being a constant torque engine. As the rpm of

a gas turbine reduces, the maximum power that it can produce drops quite

sharply. It is, of course, possible to run an engine at less than its full power

by regulating the supply of steam, fuel or electric power, depending upon

the type of engine.

The power-speed characteristics of the propeller are quite different. In a

given operating condition, the speed of the ship varies almost linearly with

propeller rpm if the wake fraction is nearly constant. This implies that the

advance coefficient J and hence the torque coefficient KQ are constant, so

that:

Pa

= = PD Tis

3

5

211' pn D KQ/TiR Tis

(11.36)

_ Le. the power required by the propeller is proportional to the cube of its revolution rate. A curve of Pa as a function of n in which Pa is proportional to n 3 is called the "propeller curve" in contrast to the curves in Fig. 11.6

349

Some },{iscellaneous Topics

100.

"

.,,

...E...OJ RQ 10 100

po. no : Maximum Engine Power end Speed Rating ,. Steam TlJrbine /

2. Electric Motor (O~C.) -" 3. Diesel Engine ..-/' 4. Gas' Turbine

\

Figure 11.6: Power-Speed Characteristics of Diffe'rent Types of Propulsion Engines. .

I.

which are "engine curves". Since engines can be run at less than full power, it would appear that the problem of engine-propeller matching has a simple solution: design the propeller so that the propeller curve intersects the engine curve at the maximum rating of the engine (PEa, no). Unfortunately, the problem is more complex as discussed in the following with respect to a diesel engine.

I

Fig. 11.7 illustrates the normal working limits of a diesel engine along with its curves of specific fuel consumption. PB is the brake power at the revolu

./

L

.


350

120 . - - - - - - - - - - - - - - - - - , . . . . . . . - - - - ,

100 t------'"'7,....c.:.....-.------=::=~~::::::1

80

60

4-0

60

100

120

Figure 11.7: Worlling Limits of a Diesel Engi,w.

tion rate n , PBO and no being the maximum continuous rating of the engine denoted by the point ~1, while sfc is the specific fuel consumption in grams per k\V'hr. There is an upper limit to the power at a given engine speed, exceeding which will cause incomplete combustion of the fuel because of in sufficient air, Line 1, or excessive cylinder pressures, Line 2. Line 3 gives the maximum power for continuous operation. Line 4 gives the upper speed limit of the engine up to which the dynamic effects on the engine parts are within acceptable limits. There is also a lower limit on the power at varying engine speeds below which the fuel supply cannot be properly regulated leading to incomplete combustion and contamination of the lubricating oil, Line 5. Fi nally, the engine should not be run at very low speeds (25-40 percent of no, Line 6) because the temperatures attained in the engine cylinders may be insufficient for ignition and the fuel combustion may be erratic. The c\!-rves of specific fuel consumption show that its minimum value occurs at about

,

r


351

90 percent of the rated engine speed and 75-85 per cent of the rated power, the specific fuel consumption at the maximum continuous rating being some 5 percent higher than the minimum.

Pe OL-_00111111!~::""":""_--~---_--l

o n' OPERATING CONDITIONS A.

BALLAST

B.

FULLY LOADED mlAL

c.

FULLY LOADED AVERAGE SERVICE

D.

FULLY LOADED, FOULED HULL, BAO WEATHER

Figure 11.8: Propeller Curves for Different Operating Conditions of a Ship.

The power~speed characteristics of the propeller (the propeller curve) de pend upon the operating conditions of the ship, as shown in Fig.U.8. The engine-propeller matching problem lies in positioning these curves in rela tion to the operating envelope of the engine. A typical solution is shown in Fig. 11.9. The propeller curve for the av~rage service condition is made

L

352

Basic Ship Propulsion 120 r---------------r-I""""lr-1

100 f - - ? t - - - - - - - - - - / - - - - : r : ENGINE MARGIN SERVICE MARGIN

80 1--+---------7'--~~'--H

60 40

40 . 60

80·

. 100

120

OPERATING CONDITlONS

\

A.

BALLAST

8.

FULLY LOADED TRIAL

C.

FULLY LOADED AVERAGE SERVICE

D.

FULLY LOADED, FOULED HULL,· BAD WEA IHER

Figure 11.9: Propeller Curves in Relation to the Engine Operating Envelope.

to pass through a point corresponding to what is termed the "continuous service rating", 80-90 percent of the maximum rated power at slightly less than 100 percent of the maximum rated rpm. The difference between the maximum c,ontinuous rating and the continuous service rating is termed the "engine margin". The difference between the power in the average service condition and the fully loaded trial condition (clean hull and propelll;!r, good weather) is the "service margin". A propeller operating on a curve to the

Some Aflscellaneous Topics

353

right of the Curve B in Fig. 11.9 is said to be running "light", and a pro peller operating on the left of Curve B is said to be running "heavy". With most marine diesel engines, it is permissible to overload the engine for short periods. e.g. 1 hour in every 12, or 2000 hours cumulative per year. The limits of permitted overloading are shown by dotted lines in Fig. 11.9. Some overspeeding during speed trials is also sometimes permitted. The maximum continuous rating of a diesel engine, i.e. the maximum power and rpm at which it can be safely run continuously for long periods, is given by the engine manufacturer for certain specified ambient conditions, e.g..air temperature 20 0 C, pressure 1 atmosphere (1.013 bar), relative hu midity 60 percent, and water temperature for cooling the turbocharged air 30 0 C. These conditions do not necessarily obtain on board a ship, and it I is necessary to l
J..----


354

- a large allowance is necessary for adverse weather conditions; - the ship is expected to spend long periods in warm water ports. A large service margin ensures that the engine can be run at high rpms without being overloaded even when the hull and propeller .become rough or the weath~r is bad. However, a larger service margin also means that a larger engine has to be selected, leading to increased machinery weight and higher initial cost. Moreover, it may not be possible to achieve the full engine power without' exceeding the rated rpm. On the other hand, with a large service margin, the operating costs of the ship are reduced because of lower fuel consumption and reduced maintenance and replacement costs. There is thus an optimum service margin for each ship that will minimise its life cycle cost. ' Example 5 A ship has a diesel engine of maximum continuous rating 6000kW at 120rpm directly connected to the propeller, which is designed to operate' in the average service condition at the continuous service rating of 85 percent maximum rated power and 95 percent rated rpm. (a) If the service margin is 20 percent, determine the propeller rpm in the fully loaded trial condition at which the ma..ximum rated power of the engine ..:ill be absorbed. (b) If due to exceptionally bad weather the power demand of the propeller increases by 40 percent over that in the trial conqition, what is the maximum rpm at which the propeller can be run if 10 percent overI'oadillg of the engine over the maximum rated torque is permitted? MaxiJItum continuous rating (mer): ~

PBO

= 6000kW

Q.o ==

P BO 21fn

=

no u'= 120 rpm = 2.05- 1 6000 21f X 2.0

=

477.465kNm (maximum rated torque)

Continuous service rat ing (csr):

PBl

n1

== .0.85 P BO 0.95710

= =

0.85

0.95

X

X

6000

2.0

=

=

1.9s- 1

--------------_ __ ..

5100kW

..

_.


355

(a) Trial Condition: 5100 PBl = -1.2 = -1.2- =

PBTrial

=

kn 3

k

=

PBTrial

!(

3

nTrial

4250 1.9 3

=

=

4250kW

619.624 kW s3

f>

L ~1

h f'

The value of n at which the maximum rated power of the engine will be absorbed is given by:

PBa k

6000 619.624

=

.

,

= 2.1314 S-l = 127.9 rpm, i.e. 106.6 %/ of no, the rated rpm.

n

!

(b) Bad Weather: PB

=

1.4

Q

=

1.lOQa

Q

=

PB 211" n

n2

= =

3.8042s- 2

X

PBTrial

=

1.4 x 619.624n 3

= 1.10 x 477.465

=

3

867.473n kW

==;' 525.212kNm

so that:

n

=

867.473n 3 211" n

1.9504s- 1

=

525.212kNm

= 117.0rpin Problems

1.

A single-screw ship has a design speed of 16.0 knots with its propeller of 4.5 m diameter running at 120 rpm. The axial and tangential components of the relative velocity of water with respect to the ship in the propeller disc at 0.7R are as follows:

e deg

o

Va m per sec: 4.5267 Vi m per sec: 0.0000

L

30 60 90 120 150 180 5.8436 7.0781 ,6.5843 6.2551 5.8436 5.3498 ~ 1.5520 -0.5702 0.4115 0.5702 0.7289 0.0000


356

where e is measured from the vertically up position. The ship may be fitted with either a four-bladed propeller ora six-bladed propeller. The four-bladed propeller of 0.9 pitch ratio has the following open water characteristics: K T

=

0.3840 - 0.2582 J - 0.1523 J2

The six-bladed propeller has istics as follows:

K T

= 0.3950 -

lOKQ

= 0.4956 - 0.2535 J - 0.2184 J2

a. pitch ratio of 0.925 and open water ~haracter

0.2707 J - 0.1575 J2

lOKQ

= 0.5031- 0.2630 J -

0.2205 J2

The point of action of thetimgential force on each blade is at a radius of 1.41 m. From the point of view of minimising unsteady propeller forces, which of the two propellers should be selected? What can you say about the harmonics present in the wake velocities? 2.

A single-screw ship has a' diesel engine of maximum continuous rating

10000 kW at 150 rpm directly connected to the propeller. The effective power

of the ship in calm water is as follows: '

Speed, knots Effective power, k\V:

15.0 3427

16.0 4324

17.0 537,8

18.0 6607

19.0 8027

I

The propeller is of 5.0 m diameter and 0.7 pitch ratio. Its open water charac teristics are as follows:

K T

=

0.5214 - 0.3953 J - 0.2528 J2

lOKQ

= 0.5745 -

0.3327 J -0.3002 J2

\

, Determine the ship speed, propeller rpm and engine power in calm water, and , in waVes which cause increases in effective power of 15, 30, 45 and 60 percent over that in calm water, given that the maximum rated torque and rpm are not to be exceeded. The wake fraction is 0.280, the thrust deduction fraction 0.250, the relative rotative efficiency 1.000 and the shafting efficiency 0.980. 3.

A ship when new has an effective power in calm water as follows: Speed, knots Effective power, kW:

15.0 2503

16.0 3178

17.0 3977

18.0 4914

19.0 6002

20.0

7256

The ship has a propeller with four blades, a diameter of 6.0 m and a pitch ratio of 0.75. \\~hen new, the propeller has a surface roughness of 30 microns and its open water characteristics are as follows:

[\"], =

0,:3360 - (1.2547 J - 0.1629 J2

10KQ

=

0.3911- 0.2266 J - 0.2044 J2

"

Some Miscellaneous Topics 1

t1

357

The width and thickness of the propeller blades at 0.75R are 1.650m and 0.085 m respectively. The main engine of the ship, directly connected to the propeller, has a maximum continuous rating of 9000 kW at 126 rpm. The ship has a wake fraction of 0.250, a thrust deduction fraction of 0.210, a relative rotative efficiency of 1.000 and a shafting efficiency of 0.980. Determine the ma."cimum speed of the ship and the corresponding propeller rpm and brake power when the ship is new. After a year in service, the effective power of the ship increases by 20 per cent, the wake fraction by 10 percent and the thrust deduction fraction by 5 percent, the relative rotative efficiency and the 'shafting efficiency remaining . unchanged. The maximum power available fro~ the engine at the rated rpm drops by 3 percent. The propeller surface rougp.ness increases to 150 microns. Determine the maximum speed of the ship in calm water after a year in ser .vice and the corresponding propeller rpm brake power if the maximum torq~e of the engine is not to be exceeded. !

ana

4. A ship is moving ahead at a speed of 16.0 knots when orders are given for the ship to stop and go astern: The engine is s~~pped immediately and started in the opposite direction after two minutes, attaining a revolution rate of 90 rpm almost at once. The ship has a displacerhent of 10000 tonnes and its added mass coefficient may be taken as 0.05. The ship's propeller is of 5.5 m diameter and 0.85 pitch ratio, and is directly connected to the engine of 8000 kW brake power at 132 rpm. The effective power of the ship is as follows: Speed, knots Effective power, kW: \

o o

5 85

10 965

15 4500

20 10919

With the ship moving ahead and the propeller reversed, the wake fraction is 0.240, the thrust deduction fraction -0.100, the relative rotative efficiency 1.000 and the shafting efficiency 0.980. The open water characteristics of the propeller for ;J. positive speed of advance and a negative revolution rate are as follows: K T = -0.300 - 0.750 J - 1.500 J2 - 00400 J3 10KQ = -0.550 - 1.058 J - 1.675 J2 - 0.317 J3

Estimate the time it takes to stop the ship and the distance that the ship tra\'els forward before stopping and reversing. Carry out the calculation at IS-second intervals. 5. The effective power of a single-screw ship iJ.1 the fully loaded trial condition (clean, newly painted hull, polished propeller, calm water) is as follows:

BllSic Ship Propulsion

358 Speed. knots Effective power, ·kW:

10.0 893

12.0 1721

14.0 2998

16.0 4848

18.0 7408

20.0

10825

The ship has a propeller of diameter 6.0m and pitch ratio 1.0 whose open water characteristics are as follows: KT = OA250 - 0.2517 J - 0.1441 J2

10KQ = 0.5994 - 0.2733 J - 0.2254 J2

The wake fraction is 0.280, the thrust deduction fraction 0.240, the relative rotative efficiency 1.040 (thrust identity) and the shafting efficiency 0.980. The propulsion plant of the ship consists of a diesel engine of 15000 kW at 118 rpm maximum continuous rating. Calculate the maximum speed of the ship in the trial condition and the corresponding brake power. In the a\"erage service condition, the effective power of the ship is 15 percent higher than in the trial conditiop., the wake fraction i~ 0.300, the thrust de duction fraction 0.250, the relative rotative efficiency 1.050 and the shafting efficiency 0.980. Calculate the maximum speed of the ship in the average service condition and the corresponding propeller rpm and brake power, and determine the "service margin" and the "engine margin" . Determine for the service condition, the maximum permissible percentage increase in effective power over that in the trial condition for the engine to run at its rated rpm without exceeding the rated torque, and the corresponding ship speed. .

;1 \1 i;

,.'i

.;

"

CHAPTER

12

Unconventional Propulsion Devices

12.1

Introduction

Propeller design aims at achieving high propulsive efficiency at low levels of vibration and noise, usually with minimum cavitation. Achieving this aim has become progressively more difficult ~th conventional propellers, i.e. the type of propellers discussed in the precec;ling chapters, as ships have become larger and faster and propeller diameters have remained limited by draught and other factors. Therefore, unconventional propulsion devices have been proposed for ships in which the performance of conventional propellers is not fully satisfactory. \

Modern unconventional propulsion devices or propulsors attempt to in crease propulsive efficiency by decreasing kinetic energy losses, and to re duce vibration, noise and cavitation by improved inflow to the propulsor. The energy losses that occur in the propulsor are associated with the axial and tangential induced velccities in the propulsor slipstream. The axial loss for a given thrust may be reduced by increasing the mass of water flowing through the propulsor per unit time and reducing the axial induced velocity. The rotational loss may be reduced by recovering the energy carried away in the slipstream by the tangential velocity. There are also drag losses and it is essential that the energy recovered from the slipstream by the various devices be greater than the energy lost due to the additional drag of these devices. 359 ,

L

Basic Ship· Propulsion

360

In addition to the hydrodynamic factors of increased propulsive efficiency and reduced vibration, ~oise and cavitation, in any proposal to use an un conventional propulsion device it is also necessary to consider the initial cost, weight and volume of the device and its associated machinery, and the reliability and maintainability of the device. The economy effected by the adoption of an unconventional propulsion device of greater efficiency com pared to a conventional propeller m~st be sufficient to recover the additional cost in a reasonable period, say five years. A number of unconventional propulsion devices have been proposed in recent times. l\1any are still in an experimental stage. On the other hand, some devices antedate the conventional propeller: the paddle wheel has a history nearly 200 years old, the contra-rotating propeller was first tried out more than 150 years ago, and a patent for waterjet propulsion was grant~d in 1661.

12.2

Paddle Wheels

A paddle wheel is a wheel that carries paddles or "floats" at its periphery and rotates about a transverse axis of the ship well above the waterline. The paddles accelerate the water and experience a reactive thrust that is trans mitted to the ship. Steamers with paddle wheels appeared at the beginning of the)9 th century but paddle wheels were gradually superseded after 1850 by scre,~ propellers for oceangoing ships. Paddle wheels are still occasionally used for, shallow draught vessels operating on inland waterways. Paddle wheels are of two types: those with fixed paddles and those with "feathering" paddles, Fig. 12.1. In the fixed paddle wheel, the paddles are rigidly attached to the wheel along radial lines. Such paddle wheels are of simple and solid construction. However, it is necessary to have a very large diameter wheel with only a small segment'of the wheel immersed in water. This ensures that the paddles enter and leave the water at a large angle to the waterline, thereby minimising the shock and the loss of energy in depressing the water at entry and elevating the water at exit. The large wheel diameter requires that the wheel run at a very low rpm, and this increases the size and weight of the propulsion machinery. In the feathering paddle wheel, the paddles are pivoted at the periphery of the wheel and attached by a

I,

l

Unconventional Propulsion Devices DIRECTION OF REVOLUTION

361 DIRECTION Of REVOLUnoN

~ ./ .". I

I

COURSE

Of

SHIP

FIXED PADDLE WHEEL

COURSE Of SHIP

FEA THERING PADDLE: WHEEL

Figure 12.1 : Paddle Wheels.

I

1 ~

mechanical linkage to an eccentric point of .the wheel such that the paddles entering and leaving the water are at a much larger angle to the waterline than the corresponding radial lines. This allows the wheel diameter to be substantially reduced (to as little as half that of a wheel with fixed paddles) and to be run at a much higher speed by a smaller propulsion plant. The efficiency of a feathering wheel is also higher than that of a fixed paddle wheel by about 10 percent and is comparable to the efficiency of a screw propeller. However, a feathering paddle wheel is heavier and costlier than a fixed paddle wheel. Moreover, the mechanical linkage system requires a high degree of maintenance. There are two paddle wheel arrangements. In "side wheelers", the paddle wheels are fitted on both sides of the ship near mid-length, so that changes in trim and the pitching of the ship have little effect on the immersion of the paddle wheels. There should preferably be a crest of the transverse wave system generated by the ship at the location of the paddle wheel. The wave wake is positive at a wave crest and the higher wake fraction increases the propulsive efficiency. Side wheelers, however, have a greater overall breadth and suffer from erratic steering while rolling in a seaway. In "stern wheelers" , the paddle wheel is fitted at the stern. Stern wheels are used in vessels such

362


.. ,

as river towboats ,\ith wide flat sterns in which draught and trim variations are small and which operate in narrow waterways. The immersion of the paddles in their lowest position varies from 0.1 to 0.8 times the height of the paddle, the larger values being used for the thinner paddles. The thrust and torque of the paddle wheel are proportional to the width of the paddle for a given height at constant wheel diameter and rpm, and for a constant ship speed. Other things being equal, the thrust and torque are also proportional to the immersed area of the wheel projected on a longitudinal plane. Design diagrams for paddle wheels have been developed by Krappinger (1954) and by Volpich and Bridge (1956, 1957, 1958). In one form of these diagrams, cur\"es representing the following functional relationship!? are plotted:

I

Q

(12.1)

pgD4

D is the diameter of the paddle wheel.

I,

Example 1 A feathering paddle wheel consists of paddles pivoted about. points on a circle of radius 2.00 m. The arm attached at right angles to each paddle is of length 0.30 m and the :link attached to this arm is on a straight line which passes through a point 0.25 m forward of the centre of the paddle wheel. The waterline is 1.8 m below the centre. Determine the angles of the paddle to the vertical when its pivot enters and leaves the surface of water. What would be the angle of the paddles entering and leaving water if the paddles were fixed instead-of feathering? In the Fig. 12.2: OA =

oC

cn =

R, radius of paddle wheel

= eR :::; eccentricity

AB = a :::; length of paddle arm

; J ,

-',

363

UncoDventional Propulsion Devices

'-'-1 R

Figure 12.2: Action of a Feathering Paddle Wheel (Example 1).

Let the angles made with the horizontal by OA, CB and AB be (), cp and respectively. Then:

AF = DE = OC + CE - OD I.e.

a cos a

= eR +

I.e.

a sin 0:

= R sin () -

R cos cp - R sin ()

R sin cp

DA= R-h i.e.

R sinO

=

R h

Therefore,

a2

l

R2

=

(e+coscp-cos())2+(sine-sincp)2

(a)

tano:

=

sin () - sin cp e + cos cp - cos ()

(b)

_

0:


364

In the present example:

R = 2.00m· eR = O.25m a = 0.30m R-h = 1.80m sinO

= R R- h =

1.80 = 0 9 2.00 .

() = 64.16° (at entry) and 115.84° (at exit)

At entry:

sinO = 0.9

cos {} = 0.4359

Substituting in (a) above:

G:~~) so that:

~

2

=

(~:~~ +cos~ _ 0.4359) 2 + (0.9 - sin~)2 .

= 62.59°

From (b): tana =

-0.01228 0.9 - sin~ = 0.1495 0.25 2.00 + cos ~ - 0.4359

= -0.0821

a = -4.70°

At eX;it:

o = 115.84°

sin{) = 0.9

cosB

= -0.4359

(~:~~r = (~:~~+COS~+0.4359)2 +(0.9-sin~)2 so that:

~

= 114.29°

tan a a

=

0.01149 0.1496

4.39"

=

= 0.0768

i.e. the angles of the paddle to the vertical when entering and leaving the water surface arc 4.70 and 4.39 degrees respectively. If the paddles were fixed, the angle of the paddles to the horizontal would be

{} = 64.16° (i.e. 25.84° to the vertical), since the paddle would be aligned along GA.

J


12.3

365

Controllable Pitch Propellers

In a controllable pitch propeller, the blades are not made integral with the boss but are mounted on separate spindles perpendicular to the propeller shaft a.xis. These spindles can be made to turn about their individual axes through a mechanism inside the boss, therehy changing the pitch of the propeller blades. Many types of pitch control mechanisms have been tried out over the years, but reliable mechanisms date back to about 1935. A typical pitch control mechanism consists of a spring loaded piston which can move a small distance forward and aft inside the propeller boss in response to hydraulic pressure transmitted to it through an oil channel in the propeller shaft. The reciprocating motion of the piston is converted into an angular motion of,the spindles by crossheads. A typical arrangement of a controllable pitch propeller is shown in Fig. 12.3. A stepless change in the pitch of the blades and hence the propeller thrust from full ahead to full astern can be made 'without changing the revolution rate or reversing the directIon of revolution of the propeller. The pitch of theI propeller may be changed from a remote location, e.g. from the navigation, bridge of the ship. Controllable pitch propellers have several advantages over conventional fixed pitch propellers:

\

/:

,

- The full power of the machinery can be utilised in all loading condi tions: static, towing and free running conditions, during ice breaking, and when the resistance of the ship increases due to weather, hull roughness, greater displacement, shallow water or other causes.

.'

- Controllable pitch propellers provide better acceleration, stopping and manoeuvring characteristics. - The propulsion plant may be operated at optimum efficiency over a range of ship speeds and displacements, even at very low speeds. - Non-reversing propulsion machinery may be used, thereby reducing its cost, weight and the space it occupies. - The speed of the ship may be varied without altering the speed of the main engine. This is useful when the 1?ain engine has a shaft driven alternator for generating electricity.

L

366

Basic Ship Propulsion 10

PROPELLER

BLADE

FLANGE SEALING

RING

8EARING

RING

1 2

BLADE TURNING RING

~ o 0

a

CRANK PIN

3

4

5 6

78

AUGNING

a

DOI'£L

0

~

.

16

~- SUDING

CROSSHEAD ~. SHOE

~. ,. PROPELLER 80SS CAP

2. SAFETY SPRING AND HOLDER PISTOt~

3. ACTUA TlNG

9. FAIRING PLATE

10. UPPER OIL TANK II. BACK PRESSURE REGULATING VALVE

4. CROSSIiEAD

12. INLET PRESSURE REGULATING VALVE

5. 80SS

1.3. PITCH CONTROL SERVOMOTOR

6

PI~CSSURE

EQUt.!...!SING GROOVES

14. INBOARD END OF SHAn

7. PROPELLER SHACT

15. POSInVE DISPLACEMENT PUMP

8. PISTON ROD

16. OIL STORAGE TANK

I

Figure 12.3: Controllable Pitch Propeller.

;,1

I I

- The speed of the ship may be directly controlled from the navigation bridge. - A controllable pitch propeller is capable of producing a higher astern thrust and at a higher efficiency. Controllable pitch propellers also

ha~e

some serious disadvantages:

- The pitch control mechanism is very complicated. (

J


367

- Controllable pitch propellers have a high initial cost, which rises sharply with diameter. - Maintenance costs are also high. - Controllable pitch propellers are highly vulnerable to damage. - The length and diameter of the propeller boss are large. - The pitch variation of the blades along the radius, optimum at the . design pitch, does not remain optimum when the pitch is changed since all the blade sections rotate through the same angle. - The blade area has to be limited to enable the pitch to be reversed, . and;this requires thicker blade sections to be used. - The efficiency of a controllable pitch propeller at its design point is lower than that of an equivalent fixed pitch propeller because of the larger boss diameter, the limited blade area and the thicker blade sec tions. When the pitch is changed from its basic design value, the efficiency falls still further because of the non-optimum pitch distribu tion. - The limited blade area and thicker blade sections result in greater cavitation and more noise. Controllable pitch propellers are used only in those ships in which the positive features of such propellers are very important, i.e. in ships which require full power operation in widely different speed or resistance ranges, which require exceptional acceleration, stopping and manoeuvring charac teristics, or which are fitted with non-reversing propulsion machinery. The types of ships in which controllable pitch propellers are often fitted include tugs, trawlers, coasters, fire floats, ferries, ice-breakers and small warships with gas turbine engines. The design of a controllable pitch propeller is carried out in a manner similar to that for conventional propellers. The design condition is selected according to the proportion of the different conditions in the operating profile of the ship. In addition to the factors considered in the design of conventional propellers, controllable pitch propeller design must also take into account the

L


368

maximum permissible spindle torque during the pitch changing operation, the limit imposed on the blade area to allow the blades to be reversed, and the higher boss diameter ratio to house the pitch changing mechanism. The boss diameter ratio of a controllable pitch propeller is usually in the range

dj D

= 0.30-0.32.

I

Example 2 A controllable pitch propeller of 4.0 m diameter has a constant pitch ratio of 0.800 at a particular setting. If the pitch is increased by turning the blades through an angle of 10 degrees, determine the resulting radial distribution of pitch and the mean pitch. The root section is at 0.3R.

= xR, the blade angle is given by tan lfJ = 271"P T = PID 71" X Initially, PolD = 0.800, and the corresponding pitch angles are lfJo. With the pitch increased, the pitch angles are lfJI = CPo + 10 degrees. At any radius

'I,'

T

x

tan lfJo

lfJo

tan lfJI

lfJI

PdD

0.30

0.8488

40.3255

50.3255

1.2056

1.1362

0.40

0.6366

32.4816

42.4816

0.9157

1.1508

0.50

0.5093

26.9896

36.9896

0.7533

1.1832

0.60

0.4244

22.9970

32.9970

0.6493

1.2240

0.70

0.3638

19.9905

29.9905

0.5771

1.2692

~

~

t

P ., !'l, ~

.

I

1 I

} I"

,\

, (

0.80

0.3183

17.6568

27.6568

0.5241

1.3171

0.90

0.2829

15.7984

25.7984

0.4834

1.3667

0.95

0.2681

15.0054

25.0054

0.4664

1.3920

1.00

0.2546

14.2866

24.2866

0.4512

1.4176

The mean pitch is calculated using Simpson's Rule as follows:

x

PI/D

xPI/D

SM

0.30

1.1362

0.3409

1

0.3409

0.30

0040

1.1508

0.4603

4

1.8413

1.60

0.50

1.1832

0.5916

2

1.1832

1.00

0.60

1.2240

0.7344

4

2.9376

2.40

f(xPI/D)

f(x)

!

I

1

Unconventional Propulsion Devices x

Pl/D

xP1/D

8M

0.70

1.2692

0.8844

2

1.7769

1.40

0.80

1.3171

1.0537

4.2147

3.20

0.90

1.3667

1.2300

4 11 2

1.8450

1.35

0.95

1.3920

1.3224

2

2.6448

1.90

1.4176

1

1.00

1.4176

Mean pitch ratio

12.4

369

=

'2

f(xPl/D)

f(x)

0.7088

0.50

17.4932

13.65

17.4932 "£f( xPl/D) -' 13.65 "£ f(x)

=

1.2816

Ducted Propellers

A ducted propeller consists of a screw propeller surrounded by a non-rotating duct (shroud or nozzle) usually in the form of an axisymmetric (annular) aerofoil with a very small gap between the propeller blade tips and the in ternal surface of the duct, Fig. 12.4. The concept of ducted propellers is due to Stipa (1931) and to Kort (1934). The ducts in ducted propellers are of two kinds: accelerating ducts which increase the inflow to the propeller, and de celerating ducts which reduce the velocity of the flow through the propeller. Accelerating ducts are often called Kort nozzles following the extensive ex perimentation carried out by Kort in developing such ducts. Decelerating ducts are sometimes termed pump jets, especially when combined with fixed blades or "stators". Accelerating ducts, Fig. 12.5(a), are used in heavily loaded propellers. The small clearance between the propeller blade tips and the duct suppresses the trailing free vortices shed by the blades, the bound vortices on the blades joining the bound vortex ring on the duct. The shape of the accelerating duct at the forward end (leading edge) increases the mass flow to the propeller. At the after end, the duct is so shaped that the cross-section increases going aft and the normal slipstream contraction is s\tppressed. The increased inflow velocity causes a decrease in the thrust and torque of the propeller. At

J.._

370


-_ It

)

FigUl'e 12.4: Ducted Propeller,

(0) ACCElERA liNG DUCT T< 1

(b) DECELERATING DUCT

T > 1

\

T

DECREASING

, ",'

Figure 12.5: Accelerating and Decelerating Ducts.


371

the same time, a circulation develops around the duct section resulting in an inward directed force which has a forward component, the duct thrust. The duct also has a drag directed aft, and this should be substantially less than the duct thrust. The total thrust of the propeller and duct taken together is then usually greater than that of an equivalent open propeller (i.e. one without a duct) whereas the torque is smaller. The efficiency of the ducted propeller is therefore greater than that of the open propeller. The improved efficiency of the ducted propeller may also be explained by the reduced kinetic energy losses in the slipstream due to the suppression of the trailing vortices and the reduction of the slipstream contraction. A decelerating duct, Fig. 12.5(b), decreases the inflow velocity into the pro pejlerthereby increasing the pressure at the'j>ropeller"Ioca:tion:Thi"Sdelays cavitation, The duct has a-cIrcUlati~~~~~~ndit'~hichp~oducesan outward directed lift with a component directed aft, Le. the duct thrust is negative. The efficiency of a ducted propeller with a decelerating duct is lower than that of an equivalent open propeller, but its cavitation properties are su perior. Decelerating ducts are therefore use,d for high speed hydrodynamic bodies in which it is necessary to minimise cavitation and underwater noise. _

"

'1

. - - _ _ _ _ . . _ . _..._._.

~

_ _----------/..-._ _

~

•.:.

An insight into the performance of ducted propellers is provided by apply ingthe a.xial momentum theory. Consider the fluid column flowing through a ducted propeller of cross-sectional area Ao, Fig. 12.6. The pressures far ahead, just ahead, just" behind and far behind the propeller are PO,Pl,P~ and P2 respectively, the propeller being regarded as an actuator disc causing an abrupt increase in pressure from Pl to P~' The velocities of the fluid with respect to the propeller far ahead, at the propeller and far behind are VA, VA + '1.'1 and VA + V2. The mass of fluid flowing through the propeller per unit time is:

m = p Ao ( VA

+ Vl )

(12.2)

The total thrust of the propeller is equal to the rate of change of momentum of the fluid in the slipstream, and is given by:

T = m [( VA

+ V2 )

= p Ao ( VA I

L

-

VA ]

+ Vl ) v2

(12.3)


372

FAR AHEAD

DUCTEO PROPELLER DISC AREA A O

FAR ASTERN

I

I

'-'-'-'

--'-'-'-'-'

p' PRESSURES 1

VELOCITIES

V A +v2 V A+v,

P2

--

P,

---

VA+v,

•

Po VA

Figure 12.6: Flow through a Ducted Propeller.

Applying the Bernoulli theorem to the sections far ahead and just ahead of the propeller, one obtains: (12.4) Similarly, for the sections far behind and just behind the propeller: (12.5) Far astern of the propeller the flow is parallel to the axis and hence there is no radial pressure gradient, so that P2 Po and:

=

(12.6) The thrust of the propeller alone is then: Tp

:=

(p; - pd Ao

:=

p Ao ( ~

+ ! V2 ) V2

(12.7)

A thrust ratio r is now defined as the ratio of the propeller thrust Tp to the total thrust T:= Tp +TD' where TD is the thrust of the duct: T

-

Tp

-

- T p +TD -

Tp

T

I

~


373

(12.8)

The thrust loading coefficient is given by: GTL

=

T

~pAo

p A ( VA

V1

+ VI ) V2

~pAo

V1 (12.9)

. This gives the following results:

1 . - [ 1 + ( 1 + 7" GT L )0.5 ] 27"

(12.10)

where VD is the induced velocity due to the duct. Thfl delivered power of the propeller is equal to the increase in the kinetic energy of the fluid per unit time: PD ~ ~m ( VA

-

+ V2 )2 -

~pAo (VA

~m

v1

+ Vd(VA + ~V2) V2

The ideal efficiency of the ducted propeller is then:

(12.11)


374

=

1 1 -v2 1+ _2_ VA

2

= Putting j5 by:

= ~ (PI -+- p~),

(12.12)

the pressure coefficient at the propeller is given

TGTL

= 1 + -2-

+

[1+(1+TGTL)O.5]2 2T

(12,13)

The effect of the drag of the duct on the efficiency of the ducted propeller can be taken into account approximately as follows: D D = ~ {J7f D l

kD =

noting that Ao

= "D 2 / 4.

vl CD

T-DD

T

Then: (12.14)

where: DD

=

duct drag:

D

=

propeller diameter;

J


375

-

duct length;

CD

-

duct drag coefficient;

kD

-

drag correction factor;

7J

-

ducted propeller efficiency including duct drag.

Eqns. (12.12), (12.13) and (12.14) yield some interesting conclusions: ;

. ~ The ideal efficiency of the ducted propeller increases as T decreases. In practice, reducing T below a certain limit would cause the flow past the inner surface of the duct to breakdown due to boundary layer separation, resulting in a sharp drop in the efficiency of the ducted propeller. - The static pressure at the propeller ihcreases· as T increases, thereby delaying cavitation. By definition, ifI T is greater than 1 the duct has a negative thrust. - The effect of duct drag reduces as the thrust loading of the ducted propeller increases. For values of GTL exceeding about 1.5 the drag of the duct is insignificant compared to the kinetic energy losses. - The effect of duct drag on efficiency can be reduced by minimising the length of the duct. In practice, however, there isa lower limit to the duct length for a given duct loading below which the flow through the duct breaks down resulting in a severe degradation of performance. The axial momentum theory only illustrates some principles and overall trends of the performance of ducted propellers. It gives no indication regard ing the actual shapes of the propeller and the duct required for generating the specified thrust; nor does it consider the effect of slipstream rotation and, tip clearance losses. A more complete treatment requires the development of a mathematical model employing singularity distributions on the propeller blades and the duct. Increasingly sophisticated theoretical analyses and design methods have been developed by, among others, Morgan and Caster (1968), Dyne (1973),

L

376


Oosterweld(1970), Ryan and Glover (1972), and Kerwin, Kinnas, Lee and Wei-Zen (1987). The design of ducted propellers is often based on the extensive model experiment data that are available. Model experiments and practical expe rience with ducted propellers offer useful design guidelines, which have been surrun~rised by Schneekluth (1987). It has been observed that the optimum diameter of a propeller in a duet is smaller than that of an equivalent open propeller, although the external diameter of the duct and the diameter of the open propeller are usually comparable. The optimum duet section profile is of an aerofoil shape, e.g. the NACA 4415 profile. However, it is possible to use a simplified profile shape, e.g. the Shushkin profile, to make the con struction of the duct easier. It is necessary to adopt a well-rounded trailing edge to obtain good astern performance, though at the cost of a small de crease in ahead performance. Typical duct section profiles are illustrated in Figs. 12.7 (a) and 12.7(b). The optimum length-diameter ratio of the duct increases with increasing thrust loading coefficient and varies from 1/ D = 0.4 to 1/D = 0.8. A smaller duct length may permit a larger propeller ~iameter depending upon the stern shape. The duct dihedral angle (the angle between the nose-tail line of the section profile and the centre line of the duct) should be such that the flow cross-section at the trailing edge is not narrowed. At the same time, the profile camber should not be so large as to give rise to boundary layer separation inside the duct. Duct dihedral angles Q = 10-15° are often used with camber ratios f /e as high as 0.05. The duct exit angle {3 should not be greater than about 2 degrees for a simplified profile and 4 degrees for an aerofoil shape. The duct cross-section is normally circular. However, in ships with very blunt \vaterlines aft, the forward part of the duct may be widened laterally, giving the forward part of the duct an elliptical cross-section, Fig. 12.7 (c). The middle and aft parts of the duct remain of circular cross-section to keep the clearance between the inner surface of the duct and the propeller blade tips small. The clearance between the duct and the propeller blades sometimes results in the jamming of the propeller in the duct due to small objects getting into the clearance space. Various measures can be taken to prevent this from happening. Fins may be fitted ahead of the duct to guide solids in the flow outside the nozzle. Several annular grooves may be provided on the inner surface of the nozzle to thicken the boundary layer and

I

,

~

I J I\

377


1 ;,

"

f-= ~~,~

LECJ

~~ci f

13 { - -

~

(b) SIMPLIfiED (SHUSHKIN) DUCT PROfiLE

DUCT PROFILE (EnlorQed)

UPTO 10°

(c) LATERAL W1bENING OF DUCT IN A BLUNT ENDED SHIP

(0) DUCT WITH AEROFOIL PROFILE

I (d) DUCT WITH GROOVES ON

(e) DUCT WITH fLATTENED

THE INNER SURF ACE

BOTTOM o ABOUT 5

f'='-\~~~;;:;:=:

SINGLE SCREW

TWIN SCREWS

(I) ALIGNMENT OF DUCTS WITH THE FLOW

Figure 12.7: Ducted Propeller Features.

, "

378


draw the solid objects towards the centre of the propeller rather than into the clearance gap, Fig. 12.7 (d). Since many of these solid objects are stones picked up from the bottom when the ship is operating in shallow water, the duct may be flattened at the bottom so that the velocity induced by the duct is reduced locally and the stones are not lifted into the flow, Fig. 12.7(e). The slight loss in efficiency that this entails may be compensated for by an increase in the dia:meter of the duct and propeller. Another problem that sometimes occurs is the drawing of air from the atmosphere into the duct leading to a loss of thrust. This is likely to occur when the leading edge of the duct is close to the surface of water and is not covered by the ship hull. The fitting of barrier plates between the atmosphere and the duct usually clears up this problem, although a better course would be to fit the duct as low down below the waterline as possible, tucked well below the stern of the ship. Cavitation may occur near the propeller blade tips and the collapse of the cavities on the adjacent duct surface may cause erosion. To deal with this, the inner surface of the duct in way of the blade tips is sometimes made of stainless steel or coated with an erosion resistant material. The duct is usually attached to the ship hull by streamlined struts. The shape of the stern is modified to facilitate the fitting of the duct and to guide the flow into it. In another arrangement the top of the duct penetrates into the hull. This allows a larger propeller diameter to be used. The wake fraction is higher resulting in an increased hull efficiency, but the flow is more itihornogeneous and propeller induced vibration may be a problem. The duct' centre line normally coincides with the propeller shaft axis, but it may be' advantageous to align the duct with the flow. In ships with twin ducted propellers, the ducts may be inclined 5 degrees inward towards the aft, while in single ducted propeller ships the duct may be inclined 5-7 degrees upwards, Fig. 12.7 (f). . The radial distribution of circulation in a ducted propeller is quite different from that in an open propeller, as shown in Fig. 12.8. The sharp drop in circulation towards the blade tips in an open propeller is not present in the ducted propeller because the trailing vortices at the tips are suppressed by the duct. Therefore, it is necessary in a ducted propeller to use a blade outline matching its different loading distribution. Propellers operating in ducts usually have wide-tipped outlines, and are known as Kaplan type

1

J

379

Unconventional Propulsion Devices OPEN PROPELLER

1 '(

r ! l'

!

~ 0.2

1.0 NORMA~

KAPLAN

MODIFIED KAPLAN

Figure 12.8: Circulation Distributi01i and Blade Outlines for Ducted Propellers..

propellers. A Kaplan type propeller has a higher efficiency than a propeller with a normal blade outline when operating in a duct. In shallow water, a Kaplan propeller has a greater likelihood of getting jammed in the duct due to stones entrained in the flow, and hence a modified Kaplan outline is sometimes adopted. Extensive methodical series data for ducted propellers have been gener ated at MARIN. The Ka propeller series in Nozzle 19A is widely used for the design of ducted propellers. Where very good astern performance is re quired, Nozzle 37 with its thicker trailing edge may be adopted. Data are also available for propellers in ducts with l/D ratios of 0.8 and 1.0, desig nated as Nozzle22 and Nozzle24 respectively. Some of these data are given in Appendix 3.

\

Ducted propellers have the following advantages over open propellers: - improved efficiency at high loading; - more homogeneous flow into the propeller and hence reduced vibration; - smaller effect 0{ loading and speed variations on efficiency;

1--


380 - improved course stabilityj

- lower vulnerability to damage due to large floating debris. The disadvantages of ducted propellers are: - poor astern propulsive performancej - reduced astern manoeuvrability and directional stabilitYj - increased susceptibility tocavitationj - greater vulnerability to damage, in shallow water due to stones being drawn into the gap between duct and propeller. Ducted propellers with accelerating ducts (Kort·nozzles) may be used with advantage in low speed vessels having high thrust loads such as tugs and trawlers. Large tankers and bulk carriers may also benefit by adopting ducted propellers. For a thrust loading coefficient CTL below 0.7, a q'ucted propeller is .less efficient than an equivalent open propeller, but fClt CTL greater than 1.5 the efficiency of a ducted propeller increases sharply above that of an open propeller. Tugs normally have a CTL greater than 3.0 while towing, and substantial improvements of as much as 30 percent in the bollard pull can be attained by using ducted propellers. For large tankers and bulk carriers of limited draught CTL is often between 2.5 and 5.0, and ducted pro pellers\should improve efficiency. In the few cases in which such ships have been fitted with ducted propellers replacing conventional propellers, savings in power 'between 5 and 12 percent have been found. However, blade tip cav-. itation when the blades are in the vertically up position and the implosion of the resulting cavities is a problem. Ducted propellers are sometimes used for steering ships. The duct is piv oted about a vertical axis, attached by a shaft (rudderstock) to the steering gear on top and supported on the solepiece at the bottom, Fig. 12.9. By turning the duct about the vertical a.xis, a lateral force is produced to steer the ship. The clearance between the duct and the propeller blade tips has to be increased slightly, at the sides to allow for the turning of the duct and at the top and bottom for bearing play. The steering duct by itself is "over balanced" , Le. it has a tendency to turn away from its position on the centre


381

Figure 12.9: Steering Duct.

line of the ship. This occurs because the centre of pressure of the duct is at about a quarter length behind the leading edge whereas the pivoting axis is at mid-length. Therefore, a vertical plate is usually fitted to the duct at its centre line just aft of the propeller. Such a plate balances the duct, improves steering and by reducing slipstream rotation improves efficiency. A steering ducted propeller has to be made smaller. than a fixed one to allow for the

L


382

rudderstock coupling on the top and the support bearing at the bottom. However, since the rudder is eliminated, the ducted propeller can be moved further aft, This allows the waterline slopes aft to be reduced and a greater distance prO\'ided between the hull and the propeller, resulting in a lower resistance and a lower thrust deduction. Decelerating ducts, as mentioned earlier, are used in high speed bodies moving in water to minimise cavitation and noise. Such ducts are usually combined' with a row of fixed fins or stator blades, the combination of duct, fins and propeller being known as a pump-jet, Fig. 12.10. The fins are used to minimise the rotation orswirl imparted to the water by the propeller. The fins are also used to support the duct. If the fins are placed ahead of the

ROTOR BLADES I

PRE-SWIRL

STATOR VANES, POST-SWIRL

SlA TOR

VANES

Figure 12.10: Pumpjet.

propeller, they are designed to impart a rotation to the flow in a direction opposite to that of the propeller. Such "pre-swirl" stators can be designed to minimise the rotation of the slipstream only at one speed. Fins fitted behind the propeller are designed to eliminate rotation in the flow leaving them.

Such "post-swirl" stators provide a low swirl in 0.11 operating conditions

and are more effective than fins placed ahead of the propeller because the

j \

I

J.\ ;:'f

.


383

velocity behind the propeller is higher. Minimisation of slipstream rotation is necessary to reduce the reaction torque on the body and prevent it spinning about its own a.xis. The close fitting duct suppresses tip vortices and allows increased tip loading in a manner similar to an accelerating duct. It also suppresses noise. The design of pump-jets may be carried out using a simplified lifting line theory, but CFD (Computational Fluid Dynamics) techniques are being in creasingly used for complex stator-rotor configurations. Decelerating duct methodical propeller series data have been produced by MARIN: Kd 5-100 propeller series in Nozzle 33. Decelerating duct propellers are widely used for torpedoes and have also been used in small,' high speed warships. Exa:mple3 A tug has a propeller of 3.0 m diameter and 0.8 pitch ratio, the open water charac teristics of which are as given in Table 4.3. The tug has a wake fraction of 0.200, a thrust deduction fraction of 0.180 in the free running condition and 0.050 in the bollard pull condition, a relative rotative efficiency of 1.000 and a shafting efficiency of 0.950. The effective power of the tug is as follows:


11.0 192.2

10.0 137.7

12.0 260.7

13.0

344.9

If the propeller is run at 150 rpm in both the free running condition and the bollard pull condition, determine the free running speed and the bollard pull, and the corresponding brake powers.

The propeller is replaced by a ducted propeller of the same diameter and 0.912 pitch ratio, whose open water characteristics are as follows: J

o

KT

0.4427 0.3497

10I
0.100 0.200 0.3957 ·0.3493 0.3467 0.3382

0.300 0.3019 0.3237

0.400 0.2518 0.3026

0.500 0.1976 0.2739

0.600 0.1376 0.2365

0.700 0.0702 0.1887

Determine the free running speed and the bollard pull and the corresponding brake powers. i

~

D

=

3.0m

w

77R

=

1.000

1Js

l___

= =

0.200

to

=

0.050 "

0.950

n = 150rpm

t 1 = 0.180

=

2.5s- 1


384 Free running condition:

RT

PE =V

=

T

RT 1-t

1 J = (l-w)V = (1- 0.200) V = 9.375 V 2.5 x 3.0 nD

KT

=

T pn 2 D4

=

T

T 1.05 x

2.5 2

x

3.04

= 518.906

10.0

11.0

12.0

13.0

5.1440

5.6584

6.1728

6.6872

PEkW

137.7

192.2

260.7

344.9

RTkN

26.679

33.967

42.234

51.576

TkN

32.645

41.423

51.504

62.898

J

0.5487

0.6036

0.6584

0.7133

KT

0.0629

0.0795

0.0993

0.1212

V{

k : ms- 1 :

Open Propeller The KT-J curves from the above table and from Table 4.3 for P/ D = 0.800 intersect at:

J = 0.6584 for which Free running speed

lOKQ

v

KT = 0.0993

= 0.1577 = 9.375 J = 9.375 x 0.6584

= 6.1728ms- 1 = 12.0 knots Delivered power,

PD = 270 pn 3 D 5 KQ 1]R

= 211" x 1.025 x 2.5

3

X

3.0

5

X

0.01577 kW 10 . 00

= 385.54kW

,, I

j

I


PD

Brake power,

7Js

385

=

385.54 0.950

= 405.8kW

Bollard pull condition: From Table 4.3 for P / D == 0.800: J{TO

= 0.3415

J{QO

=

BP = (1 - to) To

Bollard pull,

=

0.04021 at

X

=0

(1 - to ) KTO Pn5 D 4

=

(1 - 0.050)

J

0.3415

X

1.025

X

2.5 2

X

3.04 kN

168.346 kN J{QO

PD == 271"p n 5D5 -

Delivered power,

7JR

= 271" X 1.025 X 2.53 X 3.05 X 0.0~0~1 kW 1. 0 = 983.25kW Brake power,

PB

=

1035kW

Ducted Propeller Free running condition: The KT-J curve for the ducted propeller intersects the J{T-J curve from the fore going table at the same values of J and J{T as the open propeller; i.e. at the free running speed: J

=

Delivered power,

0.6584

PD

J{T

=

0.0993

0.2098

"!{Q = 271"pn 3 DC>_

7JR

= 27T" X 1.025 X 2.5 3 X 3.0""0.02098 X 1.000 kW i.

513.14kW Brake power,

l __

PB

540.15kW

386


Bollard pull condition: Bollard pull,

BP = (1 - to ) K TO P n 2 D4 = (1 0.050) x 0.4427 x 1.025 x 2.5 2 x 3.04 kN = 218.233kN

Delivered power,

PD

_ 27l"pn3 D

5 KQO

-

7JR

= 27r x 1.025

X

2.5 3 x 3.0 5

X

0.03497 1.000 kW

= 855.12kW Brake power,

PE = PD = 855.12 = 900 1 kW 7Js 0.950 .

Both the open propeller and the ducted propeller thus give the same free running speed in this example, but the ducted propeller provides a higher bollard pull at a lower power.

12.5

Supercavitating Propellers

'"\Then the d'esign conditions of a propeller are such thr.. t unacceptable levels of {avitation cannot be avoided, it is necessary to consider the use of a super:cavitating propeller. In a supercavitating propeller, a vapour filled ~ity covers the whole of the back of the propeller blade. Under proper conditions, a supercavitating propeller can provide high thrust at nearly the same efficiency as a conventional subcavitating propeller without cavitation erosion and excessive noise and vibration. Supercavitating propellers are used in ships in which high engine powers, ship speeds and propeller rpIllS are combined with small propeller diameters and low depths of immersion. /'/

For a supercmitating propeller to work efficient.ly a proper combination of the advance coefficient J and the cavitation number a are essential. The rec ommended field of application for supercavitating propellers given by Tach mindji and Morgan (1958) is shown in Fig. 12.11, and is based on achieving a propeller efficiency of at least 0.64 and eI~suring that the propellers are fully cavitating. Line 1 in the figure is based on the occurrence of incipient

387

Unconventional Propulsion Devices 2.00

\

1.50

I

",.

1.00 0.80

"'-

0.60

I

ZONE 3

~ "" ~~f -...;.

0.<-0

v w

z

N

~Q~ ~.t ~ I'-..~

w

z

~o~ 1-0

0.20

N

"~.vf

N

....... ~-

0.15

~

ZONE 1

0.10 0.08 0.06 0.04 0.2 .0.4

0.6

'0.8

1'.0

1.2

1,4

1.6

J ZONE 1

Best region for super cavitating propellers

ZONE 2

Marginal region with some cavitation on all propellers

ZONE 3

Best region for can yen tionol propell ers

ZONE 4

Region of low efficiency for all propellers

Figure 12.11 : Application of Supercavitnting Propellers.

cavitation and gives the lower limit of (Y for conventional (non-cavitating) propellers. Line 2 lepresp,nts the upper limit of (J for full cavitation to oc cur and is based on obtaining a local cavitation number at 0.7R, (JO.7R, of 0.045. If partial cavitation cannot be avoided (Zone 2) it is preferable to use propellers specially designed for such conditions, e.g. those belonging to the Gawn-Burrill Series, Gawn and Burrill (1957).


388

Supercavitating propellers were first used in racing motor boats, the de sign of such propellers being based on trial and error. Pioneering research on supercavitating propellers was carried out by Posdunine (1944), who de veloped a theoretical model of the flow, derived expressions for the thrust and efficiency of such propellers and suggested the use of wedge shaped sec tions. Tulin (1964) formulated a theory of cavity flows and this led to the development of the modern theory of supercavitating propellers. The blade section shape in supercavitating propellers differs considerably from that in conventional propellers. Since the back of the blade section is not supposed to be in contact with water in a supercavitating propeller, its design is based on ensuring· a complete separation of flow on the back at the leading edge and at the trailing edge while at the same time providing a high lift-drag ratio. Tulin developed a method for determining the shape of the camber centre line and the thickness distribution of a supercavitating foil to obtain a given lift coefficient and the optimum lift-drag ratio in two dimensional flow. This was then used in designing propellers following a method similar to that used in designing conventional propellers wi~h the circulation theory. A propeller design based on this approach requires various empirical corrections and testing in a cavitation tunnel to ensure that it satisfies the design requirements. Design charts based on this theoretical design method have been produced by Caster (1963) which cover the following ranges of design parameters: Number of blades Z

2,3 and 4

Blade area ratio, AE/Ao

0.3-0.5 (2 blades) 0.4-0.6 (3 blades) 0.4-0.7 (4 blades)

Advance coefficient, J

0-1.6

Thrust loading coefficient, GTL:

0.3-2.25

Cavitation number,

o

0"

Correction fa~tors have been provided for cavitation numbers greater than zero.

1

Unc~nventional Propulsion Devices

, \:

389

The lifting line theory in association with Tulin's linearised cavity flow theory has been used by Venning and Haberman (1962) to develop a design procedure for supercavitating propellers. Unfortunately, experiments in a cavitation tunnel have shown that this design procedure often results in propellers that have a thrust deficiency of as much as 15 percent and in which the cavity does not cover the whole of the back of the propeller bl~de. Improved design methods have been developed in which the blade sections are designed using a non-linear cavity flow theory and the effect of the cavity volume OD. the flow is taken into account. Supercavitating propeller design procedures now use vortex lattice and surface panel techniques. Initially in supercavitating propellers, aerofoil and crescent shaped (Karman-Trefftz) blade sections were tried out{ butwitlldisappointing re sults. Later, after the work of Tulin, wedg~~..il-El:ld section~J~.egaIl.j;9_b.e used. The Tulin section based on the linearised cavity flow theory has a very·sh.arp and thin leading edge, and has to be modifi~d fQr·practlc·ar use by a slight increaseIi1)hickness near the leading edge.' Ne~typ·~s·-o{biade· sections for supercavftating propellers have been developed which have thicker flat leading edges and cupped trailing edges. Such sections have been found to give better results that the Tulin sections. Blade section shapes used in supercavitating propellers are illustrated in Fig. 12.12. __~- .... 00lCQ---

INFLOW VELOCITY

ruLiN SECnON

__

~_""""l--

MODIFIED

TUUN SECnON

___-:::=-.. . .

1-

CUPPED

TRAILING EDGE

SECnON

Figure 12.12: Supercavitati1lg P"opeller Blade Sections.

I

l----

} Basic Ship Propulsion

390

Design charts for supercavitating propellers based on experiments in a cavitation tunnel have been given by Rutgersson (1979). Propellers designed on the basis of these charts perform as expected, but their efficiency is low. The open water characteristics of a series of propellers for high perfor mance craft derived from cavitation tunnel experiments have been given by Newton and Rader (1961). This Newton-Rader Series covers the following parameters: Number of blades, Z Blade area ratio, AD/Ao :

3 0.48

0.71

0.95

Pitch ratios, P/ D for the different values of AD/Ao .. ,

1.05 1.26 1.67 2.08

1.05 1.25 1.66 2.06

1.04· 1.24 1.65 2.04

Cavitation number,

0.25 to a vaJue

corresponding to

atmospheric pressure

(T

The blade section shapes used in the Newton-Rader Series are, however, not the typical wedge shapes used in supercavitating propellers, and these propellers are therefore sometimes characterised as transcavitating rather than supercavitating. However, the term transcavitating propeller is more usually applied to a propeller in which the sections at the inner radii, which have a lower resultant velocity VR , are designed ,to work in the subcavitating zone while the sections at the outer radii with a higher resultant velocity are designed for supercavitating operation. Propeller blade strength is a major problem in supercavitating propellers,

particularly because of the thin leading edge of the blade. .It is usually

necessary to use special materials such as titanium alloys, copper-beryllium

alloys and special stainless steels of high strength and hardness instead of the

nickel-copper-aluminium alloys commonly used in conventional propellers.

Supercavitating propellers have another major problem: their performance

in off-design cqnditions. A supercavitating propeller works efficiently only

when it is ,cavitating fully, i.e. at or near the design speed. Howeve~, to

reach this condition from rest, the propeller has to pass through a low speed

1

Undonventional Propulsion Devices

391

range in which it will not cavitate fully and hence work at low efficiency. As a result, unless there is a very large power margin, the vessel may never get past its low speed range. Some solutions have been proposed for deal ing with this problem, Fig. 12.13. One is to promote the development of a

VENTILATION Figure 12.13: Cavity Generation at Low Speeds.

cavity on the back of the blade at low speeds by causing flow separation through a trip wire fitted just behind the leading edge. A second method is to create a cavity artificially on the back of the propeller blade by intro ducing air through an opening in the blade, i.e. produce a cavity by ventila tion. A third method which has been adopted in some cases is to have two sets of propellers-conventional subcavitating propellers for low speed oper ation and supercavitating propellers for high speed operation. Controllable pitch supercavitating propellers and variable camber propellers, in which the blades are made in two parts one of which can be moved to alter the blade section camber, have also been used to deal with the problem of operation in off-design conditions.

\

12.6

Surface Propellers

A surface propeller or surface piercing propeller is a screw propeller which operates partly submerged in water so that the propeller blades enter and leave the surface of water c}J.c~ every revolution. Surface propellers are fitted

L

392

Basic Ship Propulsi9n

WATER LEVEL

Figure 12.14 : Surface Piercing Propeller.

just behind the hull of the ship instead of under it, Fig. 12.14. As a result, the underwater appendages required for: supporting the propeller are eliminated leading to a sharp drop in appendage resistance, which can be considerable in a small high speed vessel, as much as 30 percent of the total in some cases. At the same time, the decrease in efficiency due to the propeller not beingfully submerged is not very large. Therefore, the total power required to at~ain a given speed with a surface propeller may be substantially less than thepower required with a conycntional fully submerged propeller. Surface propellers have other advantages. They are not susceptible to cavitation since the low pressure areas on the blades are covered with air filled cavities which do not implode violently on reaching high pressure areas as water vapour filled cavities do. Since cayitation is not a problem, surface propellers may have low blade areas and hence a low drag. Also, because the surface propeller is located behind the hull and is not fully submerged, it can have a large diameter 'even when the vessel is designed to operate in shallow water. Thus, surface propellers have a good potential for use in small high speed craft of limited draught. Surface propellers also have serious disadvantages. Since the propeller blades enter and leave the water once in every revolution, the hydrodynamic forces on the blades are unsteady and, compared to, a fully submerged pro peller producing the same thrust, are also much greater in magnitude. The strength of the propeller is therefore of major concern. The periodic loading on the propeller blades also creates problems due, to fatigue and vibration. The submergence of only the lower patt of tht:<.propeller gives rise to a sig


393

nificant component of the hydrodynamic forces normal to the propeller axis, which must be withstood by the propeller shaft bearings. The unsteady pro peller torque affects the engine and the propeller shafting adversely. Surface propellers also have very poor astern performance. Finally, the submergence of a surface propeller fitted to a high speed craft may vary with speed as the craft rises in the water end trims, and this may cause the propeller to overload the engine at low speeds when the submergence is high. Ferrando (1997) suggests that the action of a surface propeller may be divided into six phases, Fig. 12.15. At high values of the advance coefficient J, a negligible amount of air enters the water with the propeller blade and

6

5

O'--_ _I...-_ _" - -_ _. L - _........-J..

°

- ' -_ _........_ _

J Figure 12.15: Phases of Action of a Surface Propeller.

is confined to the region behind the trailing edge, there being a balance between the volume of the cavity and the suction of the blade. The tip vortex of the blade is also ventilated with air. This phase is known as the "base vented" phase, 1. As the loading increases and J decreases, the suction on the back of the blade draws increasing amounts of air I and the cavity increases in size. The cavity remains confined to the region behind the trailing edge of the blade, but there may be streaks of air over the back


394

of the blade. This is the "partially vented" phase, 2. A further reduction in J results in an unstable "transition" phase, 3. The air cavity is unstable and the back of the propeller blade fluctuates between the fully wetted, partially wetted (Le. with streaks of air covering parts of the back), and the fully dry conditions. The thrust and torque coefficients, KT and KQ, also fluctuate over a significant range of values. With a still further reduction in J, the air filled cavity covers the whole of the back of the blade in the "fully vented" phase, 4. The volume of the cavity increases as J decreases.· The shapes of the KT and KQ curves depart from those of a conventional propeller because only the face of the propeller blade contributes to the thrust and torque while the back is cov~rea with air and contributes almost nothing. The air cavities attached to the blades have a large thickness and there is considerable mutual interference between the blades. As J is reduced even further, the size of the air cavities increases to such an extent that the flow between the propeller blades becomes restricted. In this "cavity blockage" phase, 5, KT and KQ decrease as J decreases. At verylow values of J, the air supply to the cavities accompanying the propeller blades 'may be blocked by excessive spray. In this "cavity choking" phase, 6, the pressure in the cavities may be lower than atmospheric, resulting in an increase in 'thrust and torque.

a:

For surface propeller of a given geometry, the thrust and torque coeffi cients are functions not only ofthe advance coefficient, the Reynolds number, the Proude number and the cavitation number, but also of the Weber num ber, Wn = V 2 LI K, where K, is the kinematic capillarity (surface tension per unit l~ngth/density) of water: (12.15)

In addition, the immersion of the propeller as a ratio of its diameter and the angle between the propeller shaft axis and the line of flow also affect surface propeller performance significantly. Since the propeller blades pierce the surface of water twice in every rev olution, surface tension is obviously an important factor in the behaviour of surface propellers. Experience has shown that the critical value of J at which transition to full ventilation occurs and there is a sharp drop in KT and K Q depends on the value of the \Veber number Wn .


395

The Froude number also influences surface propeller behaviour because such propellers operate at the surface of water and give rise to surface waves. However, model experiments seem to indicate that at sufficiently high values the effect of the Froude number vanishes, i.e. for a Froude number defined as Fn = n D g D greater than 4, there is no effect.

rJ

Surface propellers rarely suffer from cavitation. Only if high values of J, at which the air-filled cavities are small and confined to the trailing edge, are combined with very low values of the cavitation number (j is there a pos sibility of cavitation. Otherwise, the air filled cavities prevent the formation of cavities filled with water vapour which characterise cavitation. The immersion of a surface propeller has a considerable effect upon its performance, and the immersion ratio hiD, where h is the maximum tip immersion in calm water and D the propeller diameter,· greatly affects the values of KT and KQ. However, if the thrust and torque coefficients are defined in a modified form as: (12.16)

o

where A is the area of the immersed segment of the propeller disc, the effect of hiD (as separate from the effect of A appears to be negligible, particularly in the base vented and partially vented phases, 1 and 2.

o)

The angle of the propeller shaft axis to the direction of flow has a greater effect upon performance in a surface propeller than in a fully submerged propeller. The inclination of the propeller shaft axis to the relative velocity of water gives rise to an unbalanced force in a transverse plane. Such an unbalanced force is also produced by the partial submergence of the pro peller. By suitably choosing the angle of the propeller shaft in the vertical and horizontal planes, it is possible to neutralise the transverse force due to partial submergence by the transverse force due to shaft inclination. Apart from reducing the loading on the propeller shaft bearing, the elimination of the resultant transverse force also enhances propeller efficiency. There is thus an optimum shaft angle, which gives maximum propeller efficiency and zero average transverse force at the design speed of the ship. The design of surface propellers is largely based on model experiments. In carrying out model experiments, however, it must be remembered that the Froude numbe_r, the Weber number and the cavitation number cannot ,

L

396


be ignored as is usually the case in model experiments with non-cavitating fully submerged propellers. Blade section shapes used in surface propellers include wedge shaped sections, wedge shaped sections with cupped trailing edges and a patented "diamond back" shape, Fig. 12.16. The blade outline may be suitably skewed to minimise the amplitude of the unsteady forces and requce vibration.

v~

WEDGE W1lrl CUP

DIAMOND BACK

Figure 12.16: Surface Propeller Blade Section Shapes.

Like supercavitating propellers, surface propellers also have problems with operation at low speeds. Surface propellers have high pitch ratios and may have a greater immersion at speeds lower than the design speed. This may result in the engine being overloaded at low speeds. A solution to this problem is to artificially ventilate the backs of the blades by introducing air through a pipe or to direct the engine exhaust into the wake just ahead of the propeller. Alternatively, the surface propeller may be fitted to an articulated shaft by which the immersion of the propeller may be reduced if necessary to prevent overloading the engine. If the propeller shaft can be angled sideways, the surface propeller can also be used for steering and the rudder eliminated. In addition, the shaft angle can be adjusted so as to minimise the side force at each speed. Example 4 A four-bladed surface piercing propeller of 1.5 m diameter has a speed of advanc~ of 1O.Om per sec at 600rpm. The flow is along the propeller axis which is 0.1941 m

397

UncomrentionaI Propulsion Devices

above the surface of water. The thrust and torque per unit length for each blade are constant over the immersed part of the blade. and are equal to 45.00 kN per m and 14.25 kN m per m respectively. Determine the variation of the thrust, the torque and the horizontal and vertical components of the transverse force on the propeller in each revolution, and their average values. Assume that the blades are narrow.

Z=4

=

D

n

1.5m

Propeller axis above water, a

=

10.0 s-1

= 0.1941 m ~~ =

dT = 45.00 kN m- 1 dr

=

600 rpm

14.25kNmm- 1

for each blade.

If the angles with the upward vertical at which each blade enters and leaves the surface of water are 81 and 82 : 81

=

90

0

+ .

sm

-1 a

Ii. =

00'

+sm

9

_10.1941

radius at inner edge of immersed blade, ro length of immersed blade

=

~

=

a cos( 180 0

=

-

8)

=

R - ro

thrust of the blade, blade torque,

tangential force,

=

F1 =

dQ (R ro) dr

l

R

TO

,

1._

~ 7'

dQ dr = dr

dQ In R ro dr

horizontal component,

F 1H = F I cos( 180 0

-

8)

=

-F1 cos 8

vertical component,

F I v = F 1 sin( 180 0

-

8)

=

F 1 sin 8

a cos8

398


eo 105 120 135 150 165 180 195 210 225 240 255

ro

Tl

Ql

Fl

FlH

FlV

m

kN

kNm

kN

kN

kN

0.7500 0.3882 0.2745 0.2241 0.2009 0.1941 0.2009 0.2241 0.2745 0.3882 0.7500

0 16.281 21.398 23.664 24.707 25.016 24.707 23.664 21.398 16:21H 0

0 5.156 6.776 7.494 7.824 7.992 7.824 7.494 6.776 5.156

0 9.878 15.077 18.118 19.756 20.276 19.756 18.118 15.077 9.878

0 4.939 10.661 15.690 19.082 20.276 19.082 15.690 10.661 4.939

0

8.555

10.661 9.059

5.113

0

-5.113

-9.059

-10.661 -8.555

a

a

a

a

The thrust of all the blades is calculated as follows:

Ti+l(O) = Ti(O + 90°) 4

T(O) = I::1i(O) i=l

1

27r

Average T =

T(8) de

12~ dO 0

0

T1

T2

deg

kN

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 16.281 21.398 23.664

a 15 30 45 60 75 90 105 120 135 150

T4

T

kN

Ts kN

kN

kN

0.000 0.000 16.281 21.398 23.664 24.707 25.016 24.707 23.664 21.398 16.281

25.016 24.707 23.664 21.398 16.281 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

25.016 24.707 39.945 42.795 39.945 24.707 25.016 24.707 39.945 42.795 39.945

8M

f(T)

1 4 2 4 2 4 2 4 2 4 2

25.016 98.830 79.891 171.180 79.891 98.830 50.031 98.830 79.891 171.180 79.891

I

J


399

e

T1

T2

T3

T4

T

deg

leN

kN

kN

kN

kN

165 180 195 210 225 240 255 270 285 300 315 330 345 360

24.707 25.016 24.707 23 ..664 23.398 16.281 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 16.281 21.398 23.664 24.707 25.016

0.000 0.000 0.000 16.281 21.398 23.664 24.707 25.016 24.707 23.664 21.398 16.281 0.000 0.000

24.707 25.016 24.707 39.945 42.795 39.945 24.707 25.016 24.707 39.945 42.795 39.945 24.707 25.016

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

8M

4 2 4 2 4 2 4 2 4 2 4 2. 4 1

f(T)

98.830 50.031 98.830 79.891 171.180 79.891 98.830 50.031 98.830 79.891 171.180 79.891 98.830 25.016 2314.606

Average thrust, T =

! x· 15

0

x 2314.606 360 0

= 32.147kN (by Simpson's Rule)

Similarly, the variation of torque and the horizontal and vertical components of the tangential force can be 'Obtained, and their average values calculated. The results are given in the following table, and in Fig. 12.17.

e

\

deg

a 15 30 45 60 75 90 105 120 135 150 165 180

L

Q

FH

Fv

kNm

kN

kN

7.922 7.824 12.649 i3.552 12.649 7.824 7.922 7.824 12.649 13.552 12.649 7.824 7.922

20.275 19.082 20.630 21.322 20.630 19.082 20.275 19.082 20.630 21.322 20.630 19.082 20.275

0.000 -5.113 -0.504 0.000 0.504 5.113 0.000 -.5.113 -0.504 0.000 0.504 5.113 0.000

400

Basic Ship Propulsion ()

Q

FH

Fv

deg

kNm

kN

kN

195 210 225 240 255 270 285 300 315 330 345 360

7.824 12.649 13.552 12.649 7.824 7.922 7.824 12.649 13.552 12.649 7.824 7.922

19.082 20.630 21.322 20.630 19.082 20.275 19.082 20.630 21.322 20.630 19.082 20.275

-5.113 -0.504 0.000 0.504 5.113 0.000 -5.113 -0.504 0.000 0.504 5.113 0

J

The average values calculated by using Simpson's Rule to integrate the various quantities over a revolution are: Q = 10.180kNm

PH

= 20.056kN

Fv = 0

50

40 \.

20

0 10 -

T

FH

~

Q 0

:.\

:1

~~

F v

~

-10 0

e degrees

90

180

270

360

~

~

d

"

~

Figure 12.17: Loading of a SUI'face Propellel" (Example 4).

J


12.7

401

Contra-rotating Propellers

A contra-rotating propeller (set) consists of two propellers rotating in op posite directions on coaxial shafts, one propeller being placed close behind the other, Fig. 12.18. The aim is to reduce the rotational energy losses in the slipstr·eam. The first recorded use of a contra-rotating propeller is that

--..,(-

~-

Figure 12.18: Contra-rotating Propeller.

by Ericsson in 1837. Contra-rotating propellers have long been used in tor pedoes in which the mutual cancellation of the torque reactions of the two propellers prevents the torpedo from rotating about its axis and thereby imparts it directional stability. For ship propulsion, a contra-rotating propeller has the advantage that the required thrust load is distributed between two propellers so that the effici.ency is higher than with an equivalent single propeller, and the propeller diameter and blade area ratio can be reduced. However, the efficiency of a

1_

&'\..~c Ship Propulsion

402

I I t

contra-rotating propeller is less than two single propd..\e.rs producing the same total thrust. Contra-rotating propellers also ha.\"~ t'::::e disadvantages of greater weight and the complexity of the gearing and C\.."\aXial shafts. The sealing of the shafting against the ingress of water fn.~1 outside is also a major problem. In the design of contra-rotating propellers, the aft propeller is made of a slightly smaller diameter than the forward propeller tiling the slipstream contraction into account. The pitch is selected to suit the required power absorption and to ensure that the slipstream rotation ind~,""'ed by the forward propeller is cancelled by that induced by the aft propt:'il-c:'x. The circulation theory is used in the design of contra-rotating propell~"r.5, and a practical design method has been developed by Morgan (1960) ba...~ on the theory of Lerbs. This method gives good torque balance between the two propellers, a high efficiency and accurate values of the inflow velocities so that cavitatiop is better controlled. More recent design procedures are based on vortex lattice methods. Apart from their long-standing use in torpedoes, contra.-rotating propellers have been tried. out in a submarine, and more recently in a Japanese bulk carrier and a car carrier. It appears that improvements in efficiency of up to 15 percent can be obtained compared to single screws, provided favourable hull-propeller interaction ca.n be ensured. In general, however, the mechani cal complications of fitting contra-rotating propellers are not justified by the i.llcreases in efficiency that can be achieved.

12.8

Tandem Propellers

T2.1::.dem propellers consist of two propellers mounted on t he same shaft and tl.lr'ing in the same direCtion, Fig. 12.19. When a high thrust requirement is c\:>:nbined "'ith a restricted propeller diameter, there is a problem with re d.uCBd efficiency and increased cavitation. Tandem propdlers are a solution to :::.is problem since the total thrust is divided between the two propellers. Tl::.<:, propellers are usually of the same diameter and haw the same num b<:!" of blades. The two propellers must be designed h\.king into account tl:-= :nduced \'elocities due to each propeller. A set of t~ndem propellers

---------------

J

Uncom-entional Propulsion Devices

403

. Figure 12.19 : Tandem Propellers.

fitted to a British destroyer in 1900 shoWed that the aft propeller should have greater pitch than the forward propeller. Experiments indicate that at normal thrust loadings, tandem propellers have no significant advantage over equivalent single propellers. At high loadings, tandem propellers have higher efficiencies than single propellers and lower prop'eller induced vibra tion. However, tandem propellers have higher rotational energy losses, and a greater weight and cost. Semi-tandem propellers are two propellers placed one behind the other on the same shaft in which the blades· of the two propellers are skewed in opposite directions at the inner radii while at the outer radii the blades are in the same line. This arrangement is designed to reduce the effects of non-uniform inflow to the propellers, Propeller boss cap fins, Fig. 12.20, are small blades or fins fitted to the propeller boss cap, the number of fins being equal to the number of blades in the propeller. These fins weaken the hub vortex, thereby eliminating hub vortex cavitation and the noise due to the collapse of the hub vortex cavities on the rudder. Propeller boss cap fin propellers have efficiencies 3-7 percent higher than normal propellers.


404

! t

i

I Figure 12.20: Propeller Boss Cap Fin Propeller.

I,

~

12.9

Overlapping Propellers

Overlapping propellers, Fig. 12.21, consist of two propellers located at the longitudinal position of a conventional single propeller but with their shafts at a transverse separation less than the diameter of either propeller. If the two propellers are in the same transverse plane, the two shafts must be int~rlocked. If the propellers are a small distance apart along the length of the ship, their shafts may be independent. Although overlapping propellers have been investigated in detail both theoretically and experimentally they have rarely been used in practice. Overlapping propellers have the advantage of distributing the load between two propellers and therefore having a higher efficiency than an equivalent single propeller. Compared to twin screws, overlapping propellers work in a region of high wake and therefore the hull efficiency is higher. The ap pendages required to support the twin screws are also eliminated and the resistance of the ship reduced. The mutual interaction between the overlapping propellers may result in increased vibration and cavitation. It may be necessary to have the two propellers with different numbers of blades. There are also difficulties in

i

~

Il

i j : i

I· I

I

I


405

Figure 12.21 : Overlapping Propellers. .

I

supporting two propeller shafts close to each other. If the two shafts run at different speeds there may be vertical and torsional vibration. Unsteady propeller forces with overlapping propellers are greater than with a single propeller. However, the lateral forces tend to cancel each other. Overlapping propellers are usually designed to be outward turning, Le. for ahead thrust the starboard propeller turns clockwise and the port pro peller anti-clockwise when viewed from behind. The longitudinal separation between the propellers has only a small effect on efficiency though it may influence ship hull vibration. There is an optimum transverse separation between the propellers. At low separations the wake is higher. On the other hand, the aft propeller is affected by the slipstream of the forward propeller to a greater extent. The optimum transverse separation between the centre lines of overlapping propellers has been found to lie between 0.6 and 0.8 times the propeller diameter. Model experiments also show that overlapping pro pellers offer the greatest improvements in efficiency for full form ships with U-shaped afterbody sections. Substantial improvements in propulsive effi ciency of as much 20-25 percent can be obtained with overlapping propellers compared to conventional twin screws. Compared to a single screw, the gain in efficiency is small: 5-8 percent.


406

12.10

Other Multiple Propeller Arrangements

In addition to contra-rotating propellers, tandem propellers and overlapping propellers, two other multiple propeller arrangements have been proposed, In the "fore propeller" , Fig, 12.22, a small propeller is fitted just ahead of the main propeller and above its shaft on the ship centre line. This additional propeller, which is driven through a right angle drive by a vertical shaft extending from the hull above, can be made to turn about a vertical axis. .

I

\

Figure 12.22: Fore-Propeller.

This arrangement has the advantages that the additional propeller can be used for steering and for propulsion when very low ship speeds are required which are below the operating range of the main engine. The additional propeller also' increases the velocity of flow in the upper part of the main propeller disG thereby producing a more uniform flow and reducing vibration and intermittent cavitation. The disadvantages of this arrangement include the higher cost and the complex mechanism of the additional propeller, and

I I

~


407

the necessity of "feathering" the blades of the main propeller when it is idle and the ship is being propelled by the fore propeller alone. The second unconventional two-propeller arrangement consists in fitting two conventional propellers one above the other at the location of the normal single propeller, Fig. 12.23. Since the load is shared between two propellers the propeller efficiency is higher than that of a single propeller, while the higher wake at the ship centre line results in a higher hull efficiency as

i

j

k[l

ii, ~,

Figure 12.23: Two Propellers, One above the Other.

compared to twin screws. On the other hand, the diameters of the two propellers one above the other are limited and high blade areas are required to avoid cavitation. The small propeller diameters allow higher engine speeds to be used and hence lighter engines to be fitted. This arrangement of two propellers also gives better stopping ability to the ship. There may be difficulties in arranging engines and shafts one above the other, and the shafts may need to be inclined to the ship centre line. This then gives the propellers a steering capability also.

----

~

408

12.11


Vane Wheel Propellers

The vane wheel propeller, due to Otto Grim, consists of a number of narrow blades or vanes mounted on a hub just behind the propeller. The vane wheel rotates freely around the propeller shaft, which extends abaft the propeller, Fig. 12.24 (a). The' vane wheel has a larger diameter than the propeller, and the ,-anes are designed to act as turbine blades at the inner radii absorbing energy from the propeller slipstream and as propeller blades at the outer radii imparting energy to the fluid outside the slipstream. The vane wheel thus recovers some of the energy, which would otherwise be lost in the slipstream, and imparts momentum to a greater mass of fluid thereby improving the ideal efficiency of the propulsor for agiven thrust. Since a part of the thrust required is produced by the vane wheel, the loading of the propeller is reduced and its diameter may be decreased. The reduced loading also improves propeller efficiency, lowers hull pressures fluc tuations and decreases intermittent propeller cavitation. The vane wheel diameter is some 25 percent greater than the propeller diameter, but the clearance between the vanes and the hull and rudder need not be large be cause the vanes are lightly loaded. The number of vanes is larger than the .number of propeller blades: a four bladed propeller will typically have a vane wheel with nine blades. The vane wheel revolves in the same direction as the propeller and at an rpm which is 30-50 percent of the propeller rpm. Th~ pitch distribution of the vanes is such that energy is absorbed from the flow f,Lt the inner radii while at the outer radii energy is imparted to the flow. This ,is shown by the velocity diagrams in Fig. 12.24(b). The radial distri butions of circulation for the propeller and for the vane wheel are shown in Fig. 12.24(c). . The vane wheel propeller has a higher efficiency than an equivalent sin gle propeller because (a) the vane wheel provides additional thrust without absorbing additional power, (b) the propeller loading is reduced, (c) the propeller diameter may be reduced resulting in a higher wake and hence a higher hull efficiency. and (d) the removal of energy from the slipstream by the vane wheel reduces the flow velocities past the rudder, reducing its re sistance and thereby increasing the relative rotative efficiency. The smaller propeller diameter also implies a higher propeller rpm and hence a decrease in engine weight and cost. Vane wheel propellers are also less likely to suffer

,

,"

409


NINE-BLADED VANE WHEEL

FOUR-BLADED PROPELLER

(0) ARRANGEMENT

Propeller Subscripts

Vane Wheel P V

Propeller Superscripts: i Vane Wheel 0

Inner radius Ou ter Radius

(b) VELOCITY DIAGRAMS

0.02

Propeller Vane Wheel

0.01 ...

-'\,

o1-::+-:--'---'--1'---::-'-::-.........' - ' -

1.4

r

R

-0.01

(c) CIRCULATION Figu.re 12.24: Vane Wheel Propeller.

"


410

from cavitation and vibration, although the strength of the vanes may be critical in a seaway. The propeller thrust loading must be sufficiently high, Le. GTL greater than 1.0, for a significant improvement in efficiency to take place. Increases in propulsive efficiency of 7-10 percent can be achieved pro vided the designs of the propeller and the vane wheel are optimised together and the thrust loading coefficient is high enough.

12.12

Other Unconventional Screw Propellers

The use of propellers with end plates, Fig.12~25 (a), has been proposed to improve propeller efficiency by suppressing the trailing vortices shed from the propeller blade tips. Such propellers are sometimes known as TVF (tip vortex free) propellers or CLT (contracted and loaded tip) propellers. The end plates at the blade tips modify the distribution of circulation along the

(0) f?ROPELLER WITH

(b) f~ING PROPELLER

(c) PODDED PROPELLER

END PLATES

Figure 12.25: Some Unconventional Propellers.

radius and give a better spread of the trailing vortices. This decreases the induced drag of the blades and reduces tip vortex cavitation. Several differ ent types of end plates have been tried: end plates at a constant radius or aligned with the streamline at the blade tip, end plates on the suction side (back of the blade) near the leading edge and on the pressure side (face) towards the trailing edge, and end plates of different shapes - bulbous blade tips, porous tips and winglets. An improvement in efficiency up to 4' per cent can be obtained provided the blades and the end plates are optimised together to obtain the optimum distribution of circulation. Sophisticated


411

lifting surface procedures may be used for designing· propellers with end plates, but it is difficult to account for the mutual interference of the blades and end plates. There have been very few practical applications of such propellers, which may therefore be regarded as being in a developmental stage. In a ring propeller, a ring or a small duet is attached to the pro peller blade tips so that the ring revolves with the propeller, Fig. 12.25(b). The action of the ring is similar to that of the end plates: spreading tlIe trailing yortices at the tip, altering the radial distribution of circulation, reducing tip vortex cavitation, and thus improving efficiency A ring pro peller has been found to produce a greater bollard pull than a conventional propeller. The ring also increases the strength of the propeller and reduces blade vibration. In a podded propeller, the propeller is supported in a streamlined body of revolution (pod) by a vertical strut extending downward from the hull of the ship. Fig. 12.25(c). The propeller is driven through a shaft from inside the hull through bevel gears contained within the pod. The propeller with its pod and supporting strut can be rotated about a vertical axis through 360 degrees by a separate mechanism so that the propeller thrust can be directed at any angle in a horizontal plane. To allow for the rotation of the inflow to the propeller, the strut is usually not symmetrical on the two sides about its centre plane. It is possible to design the strut so that it augments the propeller thrust and compensates for the drag of the pod. The interac tion between the strut and the pod is important. Podded propellers, which are also known by several other names such as "azimuthing thrusters" and "steering rudder propellers", offer several advantages for small vessels: ex cellent manoeuvrability, very good backing performance, good speed control over the complete range and the use of non-reversing machinery. However, podded propeller units are available only upto a limited power, and the complicated Z-drive and azimuthing mechanism are serious disadvantages. There is also the possibility of interference between the podded propeller strut and the hull, or between the different podded propeller units, which are often used hi pairs. In recent years, the Z-drive has been replaced by an electric motor housed within the pod, and the power range has been continuously extended.


412

12.13

Cycloidal Propellers

A cycloidal propeller consists of a number of spade like blades fitted to a disc usually set flush with the ship hull, Fig. 12.26. The disc is made to revolve about a vertical or nearly vertical axis while the blades are made to rotate

STN 5n4..i 3'--i----rri-------L---_ _~r+_4__.::;..=!;,~

Figure 12.26: Afterbody of a Ship with Twin Cycloiclal Propellers.

about their own individual axes through a mechanical linkage system. The concept of cycloidal propellers was proposed as early as 1870, and of the two main types of such propellers that have been used in practice, the Kirsten Boeing propeller was introduced in 1928 and the Voith-Schneider propeller in 1931. In the Kirsten-Boeing propeller, the individual blades make half a rot~tion about their own axes for every revolution of the disc about the propeller axis. In the Voith-Schneider propeller the blades make one rotation about their own axes for every revolution of the disc.. With the ship moving with a speed V and the propeller revolving at an angular velocity w the path described by each blade is some form of cycloid-an epicycloid if V is less than wR, a cycloid if V is equal to wR, and a trochoid if V is greater than wR, where R is the radius of the circle described by the blades around the propeller (disc) axis. The action'ofthe Kirsten-Boeing propeller is illustrated in Fig. 12.27. Each blade revolves about the centre 0 of the propeller while rotating abput its own axis A in a manner such that the blade is aligned along the line joining

.'. .'


413

,f

f)

&

[~ f

l 7·

r.,.•.

I

~·,';

i·

.:0:

:

Figure 12.27: Action of a Kirsten-Boeing Propeller.

,i ~.'!

~

~ q:

f; " ~'l

.\

l -----

A and a point C. The resultant of the velocity of advance VA of the propeller and the tangential velocity wR of the blade produces a hydrodynamic force F which has components parallel and perpendicular to ac. By varying the position of the point C the thrust can be directed at any angle to the velocity of advance of the propeller. It may be seen from the figure that each blade undergoes half a rotation about its own axis A for each complete revolution about the centre a. In a Kirsten-Boeing propeller, the distance DC (eccentricity) is fixed and the magnitude of the thrust can only be varied for a given speed of advance by varying the angular velocity w = 21m of the propeller. The eccentricity of a vertical axis propeller is analogous to the pitch of a screw propeller. The Kirsten-Boeing propeller thus behaves like a fixed pitch screw propeller. .


414

The action of the Voith-Schneider propeller is shown in Fig. 12.28 and is similar to that of the Kirsten-Boeing propeller except that each blade makes a complete rotation about its own axis A for each revolution about the propeller axis O. The eccentricity OC can also be adjusted between zero and a value less than R, so that the magnitude of the resultant thrust of the propeller can be controlled by varying the distance OC while the

v"

(0) ZERO THRUST (Zero Angle of Attock)

(b) AHEAD THRUST

Figure 12.28: Action of a \foith-Schneider Propeller.


415

direction of the thrust can be controlled by varying the angle between OC and the velocity of advance "A. For a given propeller thrust there is an optimum combination of eccentricity OC and angular velocity w which gives the minimum delivered power. However; it is usual to run the propeller at a constant rpm and control the magnitude of thrust by controlling only the eccentricity. The Voith-Schneider propeller with its controlla?le eccentricity is thus analogous to a controllable pitch screw propeller. The Voith-Schneider propeller may be driven by an electric motor mounted above the disc on the vertical axis of the propeller! or through a horizontal shaft and a right-angled bevel gear drive by a diesel engine or electric mo tor, Fig. 12.29. The control point C is controlled by two hydraulic rams at right angles to each other so that a stepless va:riation in the magnitude and direCtioIl.'of the propeller thrust can be obt~ined. The tangential velocity

BEVEL GEARING COUPLING TO ENGINE

,.

\

ROTA T1NG DISC

BOTTOM OF SHIP

BLADES

Figlll'e 12.29: Voith-Schneider Propeller Arrangement.

of the blades is usually in the range of 10-20 m per sec, and it may be nec essary to provide reduction gearing between the engine and the propeller. For moderate speed. reductions, the right angle bevel gearing may be used for both speed reduction and turning the direction of power transmission through 90 degrees. The power transmission also includes elements such as flexible couplings to deal with misalignment's, vibration and torque fluctua

416


tions. The blades of a Voith-Schneider propeller may be made of stainless steel to withstand cavitation erosion and corrosion. Voith-Schneider propellers are usually fitted in pairs, the location of the two propellers depending upon the type of ship. In ferries, the propellers may be fitted one forward and one aft which would allow the vessel to move sideways or to turn within a circle of its own length. In a pusher tug, they may be fitted side by side at the stern, whereas in a tractor tug, the Voith Schneider propellers may be fitted side by side at the bow. The efficiency of a Voith-Schneider propeller is significantly lower than that of a conventional screw propeller. Voith-Schneider propellers are therefore fitted only in those ships in which exceptional manoeuvrability is required. The other advantages of Voith-Schneider propellers include the ability to vary the direction and magnitude of thrust without changing the speed or direction of revolution of the engine, the elimination of the rudder and steer ing gear and some simplification to the hull form of the ship. The complex mechanism requiring high maintenance is a disadvantage.

A recent innovation is a cycloidal propeller rotating about a transverse horizontal axis fitted at the stern of the ship. Such a propeller is ,known as a "Whale Tail" propeller. Example 5 A Voith-Schneider propeller has six blades each of area 0.05 m2 set on a circle of 1.0 ~ radius. The blades have lift and drag coefficients given by: CL = O.la

CD

=

0.03 + 0.1 cl

where a is the angle of attack in degrees. The propeller runs at 180 rpm and the blades are set at an eccentricity ratio of 0.50 to provide a forward thrust. The speed of advance is 5.0m per sec. Determine the thrust and torque of the propeller.

Z=6 n

A = 0.05 m 2 per blade

= 180rpm = 3.0s- 1

R = l.Om

e = 0.50

In Fig. 12.30, 0 is the centre of the propeller disc, A the centre of a blade which is oriented such that it is normal to CA. bCI R is the eccentricity ratio e. The inclinations of OA and CA with the ship centre line are e and ep respectively, where:

417


wR

r----...-,,.,t:.-----,

o

Figure 12.30: Forces

Oll

RESULTANT FORCE

e

a Voith-Schneider Propeller Blade (Example 5).

tan

sine - e RsinO - eR = cosO RcosO

The tangential velocity of the blade: wR

=

21rnR

=

21r x 3.0 x 1.0

=

18.8496ms-

1

The resultant velocity of the blade Vn makes an angle 'ljJ with the forward direction of motion. where: Vncos'ljJ Vnsin'ljJ

= -wR sinO+ VA = w R cosO

The angle of attack is: Q

= 'I/J - ( 90

+

from which CL and CD can be determined. The lift and drag of the blade are then given by: D =' CD ~pAVJ

L

_

418


I ~

~

and the thrust, torque and transverse force by:

=

T

~ = =

F

~i

L sin.,p - D L cos ( .,p

~

cos.,p

- e) + D

sin ( .,p

- e)

L cos.,p - D sin.,p

$1

The calculation is carried out in the following tables:

Ii

.~

Angles

~

e

VR sin.,p ms- 1

VR ms- 1

.,p

c¥

deg

VR cos.,p ms-l.

deg

deg

333.43 0 36.206 90.000 143.794 180.000 206.565 229.107 249.896 270.000 290.104 310.893 333.43

7.500 -1.925 -8.824 -11.350 -8.824 -1.925 7.500 16.925 23.824 26.350 23.824 16.925 7.500

18.850 16.324 9.437 0 -9.425 -16.324 -18.850 -16.324 -9.425 0 9.425 16.324 18.850

20.287 16.437 12.911 11.35 12.911 16.437 20.287 23.514 25.621 26.350 25.621 23.514 20.287

68.303 96.725 133.115 180.000 226.885 263.275 291.697 316.035 338.416 360.000 21.584 43.965 68.303

4.468

6.725

6.909 0 -6.909 -6.725 -4.868

-3.072

-1.480

0

1.480

3.072

4.868

tan
ep

deg -0.5000 0 0.7320

0 30 60 90 00 120 -0.7320 150 0 180 , 0.5000 210 1.1547 2.7320 240 00 270 300 -2.7320 330 -1.1547 '. . 360 .-0.5000

II ~

~

i ~

t

i

, I i

Forces on each blade

e

CL

deg 0 30 60 90 120 150 180

0.4868 0.6725 0.6909 0 -0.6909 -0.6725 -0.4868

CD

L

D

T

QIR

F

kN

kN

kN

kN

kN

2.424 2.318 1.175 0.099 1.175 2.318 2.424

1.372 -1.062 -2.2594 0 2.2594 1.0624 -1.372.

0.0537 5.134 0.0752 4.656 0.0777 2.951 0.0300 0 0.0777 -2.951 0.0752 -4.656 0.0537 -5.134

0.566 0.521 0.332 0.U99 0.332 0.521 0.566

4.561 4.685 2.381 0.099 2.381 4.685 4.561

I I

I I

Unconventional Propulsion Devices B

CL

CD

deg 210 240 270 300 330 360

-0.3072 -0.1480 0 0.1480 0.3072 0.4868

419

L

D

T

Q/R

F

leN

leN

kN

leN

kN

0.559 0.541 0.534 0.541 0.559 0.566

2.619 0.412 -0.534 0.412 2.619 4.561

1.739 0.900 0.534 0.900 1.739 2.424

-2.745 -2.115 0 2.115 2.745 1.372

0.0394 -4.352 0.0322 -2.489 0.0300 0 0.322 2.489 0.0394 4.352 0.0537 5.134

Totalthr ust

Ti+l(B) = 1i(B+600) !

o

T(B) = L:1i(B) i==l

Average thrust T =

dB 1o211",~(B) 2 I

1

;:....:<.....,....."..-

11" dB

B

T1

T2

Ta

T4

Ts

To

T

deg

kN

kN

kN

kN

kN

kN

kN

0 30 60 90 120 150 180 210 240 270 300 330 360

4.561 4.685 2.381 0.099 2.381 4.685 4.561 2.619 0.412 -0.534 0.412 2.619 4.561

2.381 0.099 2.381 4.685 4.561 2.619 0.412 -0.534 0.412 2.619 4.561 4.685 2.381

2.381 4.685 4.561 2.619 0.412 -0.534 0.412 2.619 4.561 4.685 2.381 0.099 2.381

4.561 2.619 0.412 -0.534 0.412 2.619 4.561 4.685 2.381 0.099 2.381 4.685 4.561

0.412 -0.534 0.412 2.619 4.561 4.685 2.381 0.099 2.381 4.685 4.561 2.619 , 0.412

0.412 2.619 4.561 4.685 2.381 0.099 2.381 4.685 4.561 2.619 0.412 -0.534 0.412

14.708 14.173 14.708 14.173 14.708 14.173 14.708 14.173 14.708 14.173 14.708 14.173 14.708


420 The average thrtb-t over each revolution:

1 = 1 2

1<

T

T dO

=

2

1<

14.352 kN

dO

evaluating the integrals by Simpson's rule. The average torqt.:e is similarly obtained as:

Q

= 8.831kNm

The-transverse force F is equal to zero when summed up over all the blades at all angles.

12.14

Waterjet Propulsion

Waterjet propulsion is said to be the oldest mechanical propulsion device considered for use in ships. Attempts to use waterjet propulsion were first made in the 1rt h century. Later, in 1775 Benjamin Franklin again proposed the use of waterjets and such a device was actually used in 1782 by James Rumsey to propel an 81-foot vessel on the River Potomac. In its present form, a waterjet propulsion unit consists of a pump inside the ship which draws water from outside, imparts an acceleration to it and discharges it in a jet ,above the waterline at the stern, the jet reaction providing the thrust to propel the ship. By directing" the jet sideways the ship can be manoeuvred, and by deflecting the jet forward an astern thrust can be obtained. \Vaterjets have many advantages over conventional screw propellers: - There are no appendages and hence there is a reduction in resistance. - Waterjet propulsion can be used in shallow water without any limita tion on the size of the pump_ - Improved manoeuvrability, stopping and backing ability are obtained. - There is no need to reverse the main engine, i.e. no reversing gear is required in the propulsion plant.


421

- The torque of the waterjet unit is constant over the complete speed range, i.e. full power can be maintained at low speeds without over loading the engine. - The speed of the ship from full ahead to full astern can be controlled without altering the rpm of the engine. - A higher static thrust can be obtained permitting high acceleration to full speed. - There is less noise and vibration. There are two important disadvantages: - The' waterjet propulsion unit occupies considerable space inside the ship, and the water passing through ,causes a significant decrease in buoyancy.

- It is necessary to provide a grating at the water inlet to prevent debris from getting in and damaging the pump. This grating decreases the efficiency of the system, particularly as it gets clogged. Waterjet propulsion is less efficient than conventional screw propulsion' at moderate speeds, but for high speed craft, waterjets may have a higher efficiency.' Vvaterjet propulsion should be considered for ships of moderate size having speeds exceeding 25 knots. Waterjet propulsion units come in a variety of designs depending upon the manufacturer. A typical unit is illustrated in Fig. 12.31. The pump is usually a mixed flow pump with an impeller comprising of a large number of wide blades with very small clearances from the casing, and stator blades before and after the impeller to minimise rotational energy losses. The pump thus has a high efficiency. The waterjet is discharged through a nozzle which can be rotated about a vertical axis to approximately 45 degrees on either side to turn the vessel. There is also a scoop or bucket which can be rotated about a horizontal axis and which can be used to deflect the waterjet downward and forward, thereby producing an astern thrust. By adjusting the position of the scoop the thrust of the waterjet can be varied from full ahead through zero to full astern without altering the discharge of the pump. '

BaBic Ship Propulsion

422

1. SWIVELLING NOZZLE 2. REVERSING SCOOP J. STEERING AND REVERSING MECHANISM 4. STATOR VANES 5. IMPELLER 6. INSPECTION HATCH 7. IMPELLER SHAFT 8. WA TER INLET Fig.ure 12.31: lVaterjet Propulsion Unit.

The waterjet propulsion unit thus provides excellent manoeuvring, stopping and reversing qualities to the vessel. There are variations in the design of waterjet propulsion units. Instead of a mixed flow pump, an axial flow pump may be used, or a radial flow pump with an axial discharge. In one design of a waterjet propulsion unit, there is a centrifugal pump with a vertical axis. The pump suction is flush with the bottom of the vessel and the discharge is through a volute casing which can be rotated through 360 degrees about the vertical pump axis. Exceptionally good manoeuvrability is obtained, particularly if t,,-in units are fitted. Waterjet propulsion may be analysed by the axial momentum theory. If m is the mass of water flowing through the waterjet system per unit time, and VJ is the exit velocity of the jet, the gross thrust is given by: Tc = mVJ

(12.17)

If the relative ,-elocity of water with respect to the ship at the w?-teriet inlet is '~4 (velocity of advance), the force required to give the ingested water


423

the speed of the ship is the momentum drag:

(12.18) The net thrust on the ship is therefore: T =. TG

-

DM = m(VJ - VA)

(12.19)

and the thrust power:

(12.20) The delivered power under ideal conditions is the increase in energy of water passing through the system per unit time:

PD = 2I m VJ2

2

(12.21)

1m (V2"":' V2 ) =

(12.22)

21 m VA

-

and the "ideal jet efficiency" is then given by:

PT 1JiJ

= -P ,= D

m(VJ - "V;.t)VA 2

J,'

A

A thrust loading coefficient is defined as: CTL =

T I' A

2P

V2 J

(12.23)

A

where A J is the area of the jet cross-section. Substituting the value of T and noting that m = p AJ V J , one obtains:

(12.24) From Eqn. (12.22):

VJ VA

2

--1

(12.25)

1JiJ

so that:

(12.26)


424 and: 7]iJ

=

4

3 + ( 1 + 2 CTL )0.5

(12.27)

The ~osses that take place in the system can be incorporated into this analysis. Some energy loss takes place at the inlet, so that the energy in the water entering the system is ~m 11'11][, where 1][ is the inlet efficiency. The water has to be raised through a height h above the undisturbed waterline, and the energy required for this purpose is m g h. Also, there are some losses in the nozzle so that the energy in the jet before these losses occur is ~m V] /7]N, where 'I7.Y is the nozzle efficiency. Therefore, the actual energy imparted to the waterjet is given by:

The jet efficiency taking these losses into account is then given by: 1])

= =

T"A, ~m V] I77N 2 (VJ

-

!m 11'17][ + mgh

VA) VA

V] I77N - 11'11][

(12.29)

+ 2gh

If the pump efficiency is 7]P, the power delivered to the pump is:

PD

=

~m V] I77N - ~m 11'17][ + mgh 7]p

(12.30)

Thepump efficiency 7]p may be regarded as having two components: the efficiency 1]PO of the pump in uniform flow ("open water") and the relative rotative efficiency 7] R: (12.31) 7]p = 7]PO 7]R If the ship has a total resistance RT at a speed V, then: PE

=

RT V

= (1 - t) T

V

--

1- w

=

7]H PT

(12.32)

where t is the thrust deduction fraction, w the wake fraction and 7]H the hull efficiency, resulting from the interaction between the hull and the waterjet unit.

:~

}

,


425

The propulsive efficiency is given by:

PE PT PJ

= - - = PT PJ PD

7]H7]J7]P07]R

(12.33)

and the Q\"erall efficiency by:

PE

7]overall

= PB =

7]H 7]J 7]PO 7]R 7]s

(12.34)

where 7]S = PD / PB is the shafting efficiency and PB the brake power of the engine. If the main engine is a gas turbine, PB should be replaced by the shaft power Ps. In the foregoing analysis, it has been assumed that the ship is propelled by a single waterjet propulsion unit. The analysis, which is based on that given by Allison (1993), follows closely the analysis of conventional screw propellers. However, this may be mislead ing, and it is necessary to define concepts such as wake fraction and thrust deduction carefully in the context of waterjet propulsion and to use appro priate values. The model testing of waterjet1propulsion systems is still in a stage of evolution and the ITTC has set up a special group to consider its various aspects. The design of waterjet systems is a highly specialised activity closely guarded by the manufacturers of such systems. One may, however, con sider a few general design features of such systems, particularly those that have an impact on the overall design of the ship. The ingestion of water from outside the hull into the waterjet system . has the effect of suction on the boundary layer around the hull and reduces its resistance. The waterjet inlet is therefore made wide laterally so that the boundary layer thickness may be reduced over a greater width. On the other hand, water ingestion also causes an increase in the relative velocity of wa ter around the hull and a decrease in pressure (or "thrust deduction"). It is therefore desirable not to locate the inlet too close to the stern where the decreased pressure would cause a large increase in resistance. Placing the inlet too far forward would increase the length of the inlet duct resulting in larger inlet losses and greater susceptibility to pump cavitation. An in creased inlet duct length would also cause a larger loss in buoyancy to the ship and occupy more internal space. The inlet duct must provide good pressure recovery and low flow distortion. This requires that the change

B8Sic Ship Propulsion

426

in direction of the flow in the duct be as small and as gradual as possible. Another important requirement is that the inlet should be located so that there is no possibility of ingestion of air. Inlets flush with the hull surface are normally used. In hydrofoil craft propelled by waterjets, the water is taken in through inlets in pods attached to the foils and led through ducts in the struts supporting the foils to the pump inside the hull. Variable area inlets may be used to match the inflow with the speed of the vessel, but fixed area inlets are more common. The design of waterjet inlets is a critical area, and sophisticated techniques of Computational Fluid Dynamics are often used for the purpose. The choice between the different types of pumps, axial flow, radial flow or i mixed flow, depends upon the "specific speed": Ns =

nQO.5 (gH)O.75

(12.35)

where n is the revolution rate, Q the discharge and H the head of the pump. Axial flow pumps have higher efficiencies at high specific speeds (0.52 and higher), whereas radial flow (centrifugal) pumps have higher effiCiencies at low specific speeds (less than about 0.25), with mixed flow pumps being most efficient in the intermediate range. For most waterjet propulsion ap plications, mixed flow pumps have the highest efficiency among the different types of pumps, although axial flow pumps have a smaller diameter. For sm~ll high speed craft, axial flow pumps may be more suitable. Where the inlet and duct losses are of such magnitude as to result in cavitation of the main 'pump impeller, an "inducer" may be provided to increase the pressure and reduce the possibility of cavitation. I

The design of a waterjet propulsion system is normally entrusted to a wa terjet system manufacturer when the machinery is to be selected and the de sign of the system finalised. The preliminary design of a waterjet propulsion unit may, however, be carried out using design charts provided by waterjet system manufacturers. A design chart of the type shown in Fig. 12.32 en ables the power re,quired to be estimated. It is necessary to apply suitable margins on the required thrust and the estimated power when selecting the machinery at the preliminary design stage, and to consider overload design conditions, i.e. displacements greater than normal, and operation in high sea states. The preliminary design of a waterjet propulsion unit can be carried


427

350 300

I I _.---- I

250 200

-

ENGINE POWER

150 100

50

T kN 0

0

60

V

Figure 12.32: Watel'jet Prop u lsion;i Preliminary Design Chart.

out and its performance estimated using performance characteristics given in charts as shown in Figs. 12.33 and 12.34. (It is important to note that Figs. 12.32, 12.33 and 12.34 are only illustrative, and are not to be used for an actual design.) Example 6 A vessel to be propelled by twin waterjets has an effective power of 2400 kW at its design s'peed of 30 knots. Carry out a preliminary design of the waterjets assuming the following values: wake fraction 0.05, thrust deduction fraction 0.03, pump effi ciency in uniform inflow 0.89, relative rotative efficiency 0.96, inlet efficiency 0.82, nozzle efficiency 0.99, shafting efficiency 0.95, and height of nozzles above water level at 30 k 0.75 m.

v =

30k

w = 0.05 77I

b

=

0.82

=

15.432ms- 1

t

= 0.03

7]N

=

0.99

PE

= 2400kW

71p = 0.89 7]8'=

0.95

= 0.96 h = 0.75m

71n


428

EO

5~ CA~TATION

70

::~

60

20~ 25

50 40

"30~

35_ . 40:

3D

20

~ o

LIMIT

'*- ,

45

50 V knots

10

kN/kW

0

l..-...-J._--'---'--l.-_-I--_'---..l._--'-_-I-_..I- 1---1

o

2000

Po2 kW/m

4000

6000

8000

10000

2

D

Fi.gllre 12.33: WC/terjet Performance Characteristics I. 250

r-----,~---r----r.--rr--r-_,_--.,._-__, 'f-) i /.1-. ,fry

•~ JI...

200

-----

I :::;

I~

/

.

/ ..... '1...

.I.~ .::)

150 5000

100

4000

~ kN/m D

50 z Ol..-._--.-J'-_---I'-_---I"- _ _l..-._---I_ _

a

10

20

30

40

50

~

_ __ . l

60

V knots

Fig/Ire 12.34: Wate/:iet Performance Characteristics II.

70

Unconventional Propulsion Devices Assume an overall propulsive efficiency, each unit is:

2400 2 x 0.530

= 2264.15 kW

PD :::: Ps Tis :::: 2264.15 x 0.950

Resistance,

RT ::::

PE

RT

T::::

T

Hence,

2400 :::: 15.432

V

80.165

2150.94

::::

PD

PD

r

f

=

.T

V :::: 30k,

From Fig. 12.32, for

::::

2(1-t)

;

i

::::

2 Tloverall

0.530. Then, the shaft power for

Delivered power,

Thrust of each unit,

I

7Joverall ::::

PE

Ps ::::

I

429

=

=

2150.94kW

155.521 kN

155.521 2 (1 - 0.03)

= 80.165 kN

0.03727kNkW- 1

=

37.27NkW- 1

= 37.27NkW- 1 :

2481.4kWm~2

f f

i . Since PD :::: 2150.94kW,

D

2

::::

2150.94 2 2481.4 :::: 0.8668m D:::: 0.9310m

The thrust at full power as a function of ship speed is obtained from Fig. 12.33 as given in the following. For

~~

:::: 2481.4kWm- 2:

F knots

15

T/D 2 kNm- 2

:

TkN 2( 1- t) TkN:

111.92

20 105.18

25

30

35

99.108

92.484

86.827

97.012

91.172

85.907

80.165

75.261

188.203

176.874

166.660

155.520

146.006

Fig. 12.33 also indicates that full power should not be continuously maintained below about 22 knots (Limit 1).

Assumi.ng that

Nozzle Area == 0.40 Inlet Area


430 Nozzle area (jet area)

AJ = 0.40 x

To

4" D 2

= 0.40 x 4"To

x 0.9310

2

= 0.2723m 2

If the velocity of the jet is VJ:

Mass flux,

Speed of advance,

VA

=

(1 - w) V

= (1 -

0.05) x 15.432

= 14.6604ms- 1

Thrust,

T

Le.

80.165

= m ( VJ - VA) = p AJ VJ ( VJ - VA )

= 1.025 x 0.2723 VJ (VJ - 14.6604)

so that

VJ = 25.7956ms- 1

and

m = 1.025 x 0.2723 x 25.7956 = 7.1997tonness- 1

,

The hull efficiency,

7JH =

1-t

1-w

=

1- 0.03 1 - 0.05

=

1.0211

The jet efficiency is given by:

1JJ

= =

~m[VJ/7JN - V~771 +2gh] ~ x 7.1997

80.165 x 14.6604 - 14.66042 x 0.82 + 2 x 9.81 x 0.75 J

[25.7956 2 /0.99

0.6394

The overall propulsive efficiency is then:

1Joverall

= =

1JH 1Jp 1JR'7J 1J8

=

0.5299

1.0211 x 0.89 x 0.96 x 0.6394 x 0.95

~ ! t.. iI",


431

This is very close to the initially assumed value. If there had been a significant difference between the initial and final values, the calculations would have to be iterated until a satisfactory agreement was obtained.

12.15

Flow Improvement Devices

The propeller in a single screw ship works in an extremely complex flow. There is the boundary layer on the hull which grows in thickness from for ward to aft and within which the velocity rises sharply from zero to ap proximately the free stream velocity (the velocity determined on the basis of inviscid flow). In many ships, there may be boundary layer separation resulting in eddies being carried to the propeller. lThe flow at the propeller is not usually.·parallel to its axis and there may be! significant ~rossflow veloci ties normal to the axis. The ship may shed bilge vortices. The non-uniform flow resulting from all these factors causes a reduction in the efficiency of the propeller and tends to increase intermittent cavitation and propeller induced vibration. The undesirable effects of the complex flow at the stern of a ship can be mitigated to some extent by fitting devices to improve the nature of the flow into the propeller. In addition to improving the flow, some of these devices also directly improve the efficiency of the propeller. The asymmetric stern, in which the transverse sections at the after end of the ship .are not symmetrical port and starboard, is designed to impart a swirl to the flow just ahead of the propeller to counter the rotation of the flow induced by the propeller, Fig. 12.35. The rotational energy lost in the propeller slipstream is thereby reduced. The asymmetric stern produces a smaller mean wake fraction and a lower turbulence level but a somewhat greater non-uniformity in the flow. The obvious disadvantage of an asym metric stern is the i l1 creased difficulty in its construction and the higher cost. An improvement in propulsive efficiency of 6-7 percent may be obtained by using this type of stern instead of a conventional stern. The object of imparting a rotation to the flow opposite to that caused by the propeller can also be achieved by fitting an array of fixed blades radiating outwards just ahead of the propeller. Such a "pre-swirl" stator increases the relative tangential velocity of the blades. The propeller can be provided a more uniform radial distribution of circulation, the circulation

l- - - - - - - - - - - - - - - -

Ba.sic Ship Propulsion

432

Figur'e 12,35: Asymmetric Stern.

on the stator blades being opposite to that on the propeller blades. The tips of the stator blades should be at a radius some 15 percent greater than the radius of the propeller to ensure that the tip vortices shed by the stator blades pass clear of the propeller. The number of fixed blades and the number of propeller blades should be such that their least common multiple is l~rge) e.g. 5 propeller blades and 9 stator blades. It may also be necessary to arrange the stator blades non-uniformly for reducing vibration. Fixed blades may also be fitted behind the propeller. Such a "post-swirl" stator is designed to reco\'er the rotational energy in the propeller slipstream and convert it into a forward thrust. A typical pre-swirl stator is the Mitsubishi reaction fin system) Fig. 12.36. In addition to providing a pre-swirl to cancel the slipstream rotation of the propeller, the fins are supposed to reduce the turbulence in the flow and decrease noise and vibration. In some cases, a ring at the fin tips is fitted to provide support. The fins improve the flow at the inner radii of the propeller and eliminate blade root erosion. The improved flow to the propeller also produces better hull-propeller interaction. The fins have a radially varying pitch such that a fonvurd thrust is produced. A 4-8 percent

1

~_J

,


433

I Figure 12.36: Mitsubishi Reaction Fin System.

Figure 12.37: Grot/wes Spoilers.

saving in power may be obtained by fitting the Mitsubishi reaction fin, the effect being greater in the ballast condition. Grothues spoilers, named after their inventor Grothues-Spork, consist of a set of triangular fins attached to the hull just ahead of the propeller, Fig. 12.37. The fins are designed to improve the flow into the propeller by conve'rting the vertical component of the flow due to bilge vortices into a horizontal flow, and thus recover energy. The fins produce a small forward thrust, increase the mass flow through the propeller disc and reduce angular velocity variations in the flow. Grothues spoilers thereby reduce resistance,

L

434


improve propeller efficiency and reduce vibration. The shape, position and number of fins must be determined by model tests. Power savings ranging from about 3 percent for fine vessels with low breadth-draught ratios to about 9 percent for full tankers are possible using Grothues spoilers, greater improvements being achieved in the ballast condition.

!:

Figure 12.38: Mitsui Integrated Duct.

The Mitsui integrated duct is a duct located just forward of the propeller and integrated into the ship hull, Fig. 12.38. The duet is asymmetric port and starboard and the profile chord is larger at the top than at the bottom. The ttailing edge of the duct is aligned with the propeller blade tips. The in tegrated duet stabilises and homogenises the flow into the propeller. There is also a small decrease in the resistance of the ship. The duet increases the mass of water flowing through the propeller per unit time, and there fore increases the propeller efficiency. The thrust deduction is also reduced. The improvement in the flow to the propeller causes a slight reduction in cavitation and hull pressure fluctuation. There is also said to be some im provement in manoeuvrability due to the integrated duct. Owing to the inward direction of the flow at the stern the duct produces a forward thrust. The power saving that can be obtained by fitting a Mitsui integrated duct is about 5-10 percent, the greater values being obtained for full form.ships.

J

~


435

The wake equalising duct, due to Schneekluth, is a 'duct attached to the hull ahead of the propeller in the upper part, the diameter of the duct being about half the propeller diameter, Fig. 12.39. The duct may be in the form of a complete ring or as two halves on either side of the hull. ships with . .

In

WAKE EQUALISING

DUCT

I f.

l f.

~

.'

(

f

Figure 12.39: Schneelduth Walle Equalising Duct.

an asymmetric stern or in high speed ships the duct may be fitted only on one side. The wake equalising duct is sometimes used in combination with Grothues spoilers. The main function of the duct is to accelerate the flow in the upper half of the propeller disc where it is normally slower than in the lower half. This increases the amount of water flowing through the propeller disc. The advantages that a wake equalising duct can give include: - Increase in propeller efficiency due to a greater mass flow through the propeller disc. - Reduction in flow separation at the stern resulting in reduced ship resistance. - Generation of forward thrust by the duct. - Reduction in unsteady propeller forces due to a more uniform inflow. - Reduced cavitation due to lower propeller loading resulting from the increased mass flow and also because of more uniform flow.

l

----


436

- Improved manoeuvrability through better flow to the rudder, and bet

ter course-keeping due to the lateral area of the duct.

The design parameters of the wake equalising duct, viz. diameter, profile shape and length, duct dihedral angle (cone angle), and the angle of the duct axis with respect to horizontal and vertical planes, may be determined by comparison with existing installations or by model tests. Model experiments indicate that a saving in power of as much as 15 percent may be obtained by fitting a Schneekluth duct, although the gains realised in full size ships are somewhat smaller. The main advantages of the duct are a reduction in resistance and an improvement in propulsive efficiency. The greatest benefits are obtained in ships with block coefficients of 0.60 and more, propeller diameters exceeding 2.20 m, angles of run at 0.75R above the propeller axis greater than 10 degrees, and speeds in the range 12-18 knots. The benefits of the wake equalising duct are greater in the fully loaded condition than in the ballast condition with the vessel trimmed aft. In high speed ships, the advantages of the duct lie mostly in its producing more uniform flow, thereby reducing propeller induced vibration and cavitation.

12.16

Design Approach

The detailed design of unconventional propulsion devices must take into ac count the nature of the flow at the stern of a ship and the interaction between the propeller and its inflow. Various computational methods ranging from potential flow-based lifting line methods to viscous flow RANS solvers are being tried out for the design of such devices. The agreement between the results of CFD codes and model experiments is not satisfactory in many cases. Nevertheless, CFD codes are being used for the practical design of complex hull-propulsor configurations. On the other hand, lifting line and lifting surface techniques are often adequate for the design of unconventional propellers and flow improvement devices. Generally, however, model test ing is the most reliable method for the design optimisation of unconventional propulsion devices at present. Scale effects are a major problem, and the im provements in efficiency predicted on the basis of model tests for a particular device are often found to be somewhat optimistic in actual practice.

J


437

Problems 1.

A feathering paddle wheel consists of paddles pivoted about points on a circle ofraclius 2.5 m. The arm attached at right angles to each paddle is 0.35 m long. The waterline is 2.0 m below the centre of the paddle wheel. Determine the distance from the wheel centre of the point through which the link attached to the paddle arm must pass if the paddle is to have an angle of 10 degrees to the vertical when its pivot enters water. What would be the diameter of a paddle wheel with fixed paddles to have the same angle?

2. The open water characteristics of a controllable pitch propeller of 3.0 m diam eter are given by:

KT = (-0.077

~: + 0.546 ~ -

0.054) + (0.289

~: -

0',485

~-

0.050) . J

P ) p2 + ( -0.252 D2 + 0.558 D - 0.447 J2

10KQ = (0.522

~: + 0.072 ~ + 0.021) + (0.044 ~: -

+ (0.067

~:

- 0.199

~-

0.287

~-

0.038) J

0.099) J2

This propeller is fitted in a tug which has an engine of 1035 kW brake power at 600 rpm connected to the propeller through 4:1 reduction gearing. The effective power PE of the tug at different speeds VK in knots is as follows: VK, knots: PE,kW

:

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0

0.49

5.57

23.04

63.07

137.72

260.69

447.13

713.52

The wake fraction is 0.200, the relative rotative efficiency is 1.000, the thrust deduction fraction is given by t = 0.05+0.01083 VJ(, and the shafting efficiency is 0.950. Determine how the pitch ratio must be varied with speed to ensure full power absorption. Calculate the towrope pull at the different speeds. What is the free running speed of the tug? 3. A tug is fitted with a ducted propeller of 3.0 m diameter and 0.973 pitch ratio, the open water characteristics of which are as follows:

I

l---------


438

J

0

0.200

0.400

0.600

0.800

1.000

KT

0.4862

0.3897

0.2902

0.1752

0.0320

-0.1522

lOKQ:

0.4021

0.3891

0.'3539

0.2866

0.2196

0.0152

KTD:

0.2469

0.1587

0.0885

0.0223

-0.0724

-0.2232

KTD is the duct thrust coefficient, KTD = TDlpn 2 D 4 •

The tug has a diesel engine of 1035 kW brake power at 600 rpm connected to the propeller through 4:1 reduction gearing. The wake fraction is 0.200, the thrust deduction fraction is given by t = 0.050 + 0.01083 VK where VK is the speed of the vessel in 'knots, the relative rotative efficiency is 1.000 and the shafting efficiency 0.950. The effective power of the tug is as follows: Speed, knots Effective power, kW :

10.0

11.0

12.0

13.0

137.72

192.25

260.69

344.98

Assuming that the maximum torque and rpm of the engine are not to be exceeded, determine the bollard pull and the free running speed· of the tug, and the corresponding engine powers and rpms. Calculate also the duct thrust in the two conditions. 4.

In a shallow draught vessel with two surface piercing propellers, each propeller has four blades, a diameter of 1.5 m and a pitch ratio of 1.0. At the design speed, the propellers run at 600 rpm and the relative velocity of water with respect to each propeller is 5.0 m per sec in a horizontal direction parallel to the centre line of the vessel. The propeller axes are horizontal but inclined at 10 degrees outward towards aft to the vessel centre line, and are 0.1941 m . above the waterline. For the section at 0.7R of each propeller, the lift and , drag coefficients per degree angle of attack are 0.015 and 0.001 respectively, and the section chord is 0.3 m. Assuming that the axial and tangential forces per unit immersed blade length are constant and equal to the values at 0.7R, determine the thrust, the torque and the horizontal and vertical components of the force normal to the axis of each propeller. If the wake fraction is 0.05 and the thrust deduction fraction 0.02, what is the effective power of the vessel? Assume that the propeller blades are narrow and that the induced velocities can be neglected.

5. A tug is fitted with two Voith-Schneider propellers, each consisting of six blades each of area 0.05 m 2 set on a circle of 1.0 m radius. The blades have lift and drag coefficients given by: CL = 0.10:

CD = 0.03 + o.lcI


439

where Q is the angle of attack in degrees. Each propeller runs at 180 rpm and the blades are set at an eccentricity ratio of 0.20. Determine the bollard pull and the delivered power" assuming that there is no thrust deduction. 6. A planing craft is propelled by a single waterjet unit of 3000 kW shaft power. The inlet diameter is 1.0 m and the nozzle diameter is 0.6325 m. The effective power of the planing craft is as follows: Speed

Effective power

Speed

Effective power

knots

kW

knots

kW

15.0 20.0 25.0 30.0

47.9 148.5 439.1 1232.4

35.0 40.0 45.0 50.0 "

1548.3 1897.5 2280.6 2697.3

The wake fraction and the thrust deduction fraction relevant to the waterjet unit are 0.060 and 0.020 respectively. The waterjet nozzle is 0.6 m above the waterline at full power. The shafting efficiency is 0.970, the nozzle efficiency 0.985 and the waterjet inlet efficiency 0.850. Using Fig. 12.33, determine the speed of the craft at full power, and the waterjet pump efficiency as installed.

ApPENDIX

1

Some Properties of Air and Water Density: Density P in kg per m 3 ; temperature t in degrees Celsius. Air (at normal atmospheric pressure)

1.

Pair

2.

353.172 = 273.16 + t

. (Al.I)

Fresh water \

PF\V

= 999.9227 + 0.4671409

t 1O

- 0.6834635

(ItO)

2

+ 0.01943808 (ItO) 3 3.

(A1.2)

Sea water (3.5 percent salinity) psw = 1028.1474 - 0.5547975

t - 0.6344066 (ItO) 2 1O

+ 0.03629289 (ItO) 3

(A1.3)

440

J

Appendi.x 1

441

Kinematic Viscosity: Kinematic visco.sity

1/

in m 2 per sec.

(The following formulas are simplifications of those given in the ITTC 1978 performance prediction method.)

1.

Fresh water

10 2.

6

1/FW

= 0.000585 t 2

-

0.04765 t

+ 1.72256

(AlA)

Sea water (3.5 percent salinity)

10 6 1/BW

= 0.000695 t 2

-

0.052078 t

+ 1.820219

(A1.5)

Vapour Pressure: Vapour pressure PV in kN per m 2 ;

1.

Fresh water PV .

=

0.6108 + 0.4442 (:0)

+ 0.0~1414 (ItO) 2 + 0.02788 (ItO) 3

t

+ 0.002474 (ItO) 4 + 0.03007(l O) 5

(A1.6)'

The vapour pressure of sea water is about 3.3 percent less than the vapour pressure of fresh water at the same temperature.

b

ApPENDIX

2

Aerofoil Sections used in Marine Propellers Source: Abbott and Doenhoff (1959)

: .. ,

TableA2.1

Geometry Mean Lines ?-J.ACA NACA a = 0.8 a = 1.0 (modified)

Thickness Distributions NACA-16 NACA-66 (modified)

x/c

Yc(x)/f

Yc(x)/ f

Yt(x)/t

Yt(x)/t

0 0.0125 0.0250 0.0500 0.0750 0.1000 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000

0 0.0907 0.1586 0.2711 0.3657 0.4482 0.6993 0.8633 0.9614 1.0000 0.9785 0.8890 0.7026

0 0.097 0.169 0.287 0.384 0.469 0.722 0.881 0.971 1.000 0.971 0.881 0.722

0 0.1077 0.1504 0.2091 0.2527 0.2881 0.3887 0.4514 0.4879 0.5000 0.4862 0.4391 0.3499

0 0.1044

0.1466

0.2066

0.2525

0.2907

0.4000

0.4637

0.4952

0.4962

0.4653

0.4035

0.3110

\.

\

:t

442

\;

_--A

Appendi.x 2

443 Table A2.1 (Contd.) Mean Lines NACA NACA a = 1.0 a = 0.8 (modified)

Thickness Distributions NACA-16 NACA-66 (modified)

x/c

Yc(x)/ f

Yc(x)/ f

Yt(x)/t

Yt(x)/t

0.9000 1'.0000

0.3687

0.469

o

o

0.2098 0.0100

0.1877 0.0333

x

-

distance from leading edge along nose-taillinej

c

=

section chord;

t

section thickness;

f

section camber;

Yc = ordinate of mean line from nose~tail line; Yt = ordinate of section face (and back) from the mean line. For the NACA-66 (modified) thickness distribution, Yt(x) /t x/c = 0.4500.

= 0.5000 at

The leading edge radii of the two sections are as follows: NACA-16: radius = 0.4885 t 2 /c

l

NACA-66: radius (modified)

= 0.448t 2 /c

ApPENDIX

3

Propeller Methodical Series Data (a) B-Series (Troost, Wageningen, MARIN)

Geometry Nomenclature

c

Blade section chord

D

Propeller diameter

r

Blade section radius

R

Propeller radius

t,

1Iaximum thickness of blade section

to

Blade thickness extrapolated to zero radius

tl

Blade thickness at blade tip

tLE

Thickness at leading edge (before rounding)

tTE

Thickness at trailing edge (before rounding)

x

Distance from leading edge of blade section

Xo

Distance of leading edge from propeller reference line

\

X

Distance of position of maximum thickness from leading edge

m

Yf

Ordinate of face of blade section at x

Yb

Ordinate of back of blade section at x

I, REFERENCE BLADE

I

UNE

y

,

'

r------,... X -------A" r-------c------_,/" Figure A3.1 : Blade Section Geometry of B Series Propellers.

TableA3.1 Blade Thickness

2

3

4

5

6

7

tolD:

0.055

0.050

0.045

0.040

0.035

0.030

tdD:

0.003

0.003

0.003

0.003

0.003

0.003

Z

t D

-

l ------

=

to D

1) (to - _ .tR D D

T ---

(A3.1)


446

~.

'\

~'f

TableA3.2

;,

Blade Outline for B-Series C

=

D

AE/Ao Cl

Cl

0.2 0.3 0.4 0.4 0.6 0.7 0.8 0.9 1.0

1.633 1.832 2.000 2.120 2.186 2.168 2.127 1.657 0

xo/c

(A3.2)

Z

Z = 4,5,6,7

Z = 2,3

r/R

>,'.

xm/c

0.616 0.350 0.611 . 0.350 0.599 0.350 0.583 0.355 0.558 0.389 0.526 0.442 0.481 0.478 0.400 0.500

Cl

xo/c

xm/c

1.662 1.882 2.050 2.152 2.187 2.144 1.970 1.582 0

0.617 0.613 0.601 0.586 0.561 0.524 0.463 0.351

0.350 0.350 0.350 0.350 0.389 0.443 0.479 0.500

.~

At MARIN, a modified B-Series has been developed that has a blade outline which is wider towards the tip and narrower towards the root, and thus has better cavitation properties. The open water characteristics of this modified series, known as the BB-Series, are said to be identical to those of the B-Series. The particulars of the blade outline of the BB-Series are given in Table A3.3. Table A3.3 Blade Outline for BB-Series

r/R

Blade Outline of BB-Series Cl

xo/c

xm/c

0.200

1.600

0.581

0.350

0.300

1.832

0.584

0.350

0.400

2.023

0.580

0.351

0.500

2.163

0.570

0.355

~

_________________J.

Appendix 3

447 TableA3.3 (Contd.)

Blade Outline of BB-Series

r/R

Cl

xo/c

xm/c

0.600

2.243

0.552

0.389

0.700

2.247

0.524

0.443

0.800

2.132

0.480

0.486

0.850

2.005

0.448

0.498

0.900

1.798

0.402

0.500

0.950

1.434

0.318

0.500

0.975

1.122

0.227

0.500

1.000

0

Blade Section Face and Back Ordinates For x :S

= v (t

t LE )

Yb = v (t

tr E)

Vb

Xm

(A3.3) For x In :S x :S c

YI = u (t trE)

where u and v are given in TableA3.4(a-d)

K r , KQ Polynomials

(A3.4)

These polynomials are given for a Reynolds number of 2 x 10 6 • The coef ficients Cr and CQ are given in Tables A3.5 and A3.6 respectively.

l ----

~ ~

TableA3.4

(Xl

Blade Section Ordinates (a)

-

xlxm

0.05

()

rill.

.--_.

0.10

o.n

0.20

O.:W

---_.------- .. ---_._-

0.'10

O.f>O

(). GO ~-_

n,R,)

] .00

.. _-_._._------

Values of 'U

0.2

0.35GO

0.2821

0.2353

0.2000

0.1685

0.1180

0.0804

0.0520

0.0304

0.0049

0

0.3

0.2923

0.2186

0.1760

0.1445

0.1191

0.0790

0.0503

0.0300

0.0148

0.0027

0

0.4

0.2131

0.1467

0.1088

0.0833

0.0637

0.0357

0.0189

0.0090

0.0033

0

0

0,5

0.1278

0.0778

0.0500

0.0328

0.0211

0.0085

0.0034

0.0008

0

0

0

0.6

0.0382

0.0169

0.0067

0.0022

0.0006

a

0

0

0

0

0

0.7-1.0

0

0

0

0

0

0

0

0

0

0

0

tl:l ~

n' ~ '0'

1 c til

o'

::l

,

L_._,.

-_,e.

c __

,~·~-

_ _ ........

_~.,.~

. _••

~._~ ~~_...

• ._.

-

'.';·.-.:'r
..,...,. co

,j::>,

TableA3.4

en

0

/'

Blade Section Ordinates

(c)

0

0.05

0.10

0.15

._----_ ... -

O.3D

0.20

rjR

xjx m 0.40

0.50

O.GO

0.80

1.00

Values of v

0.2

0.3560

0.4381

0.5193

0.5905

0.6462

0.7370

0.8081

0.8690

0.9179

0.9799

1.0000

0.3

0.2293

0.4076

0.4957

0.5710

0.6321

0.7295

0.8023 '0.8615

0.9068

0.9777

1.0000

0.4

0.2181

0.3402

0.4323

0.5168

0.5857

0.6947

0.7782 ' 0.8435

0.8966

0.9725

1.0000

0.5

0.1278

0.2528

0.3556

0.4463

0.5250

0.6515

0.7512

0.8283

0.8880

0.9710

1.0000

0.6

0.0382

0.1654

0.2787

0.3797

0.4626

0.6060

0.7200

0.8090

0.8790

0.9690

1.0000

0.7

0

0.1240

0.2337

0.3300

0.4140

0.5615

0.6840

0.7850

0.8660

0.9675

1.0000

0.8

0

0.1050

0.2028

0.2925

0.3765

0.5265

0.6545

0.7635

0.8520

0.9635

1.0000

0.9

0

0.0975

0.1900

0.2775

0.3600

0.5100

0.6400

0.7500

0.8400

0.9600

1.0000

tl:l

-. ~

(")

-.

~ '0

1 >::

-.

iii 0

::J

L--------_.-----.. -----_.. .

,~,~.

"~/.-':'
",,. ..... :

~;~~{._'~:'t:""='~~"lIS'~

TableA3.4

:g>

Blade Section Ordinates

:;,

Cll

Q.

r:' ;..:

(d)

V,:)

(x xm)/(c x m ) 0

0.20

0.40

0.50

r/R

0.60

0.70

0.80

0.90

0.95

1.00

Values of v

0.2

1.0000

0.9618

0.8576

0.7875

0.7049

0.6105

0.5027

0.3855

0.3270

0.2826

0.3

1.0000

0.9616

0.8467

0.7711

0.6818

0.5828

0.4693

0.3460

0.2840

0.2306

0.4

1.0000

0.9645

0.8459

0.7641

0.6567

0.5435

0.4130

0.2782

0.2105

0.1467

0.5

1.0000

0.9639

0.8456

0.7580

0.6451

0.5240

0.3759

0.2195

0.1370

0.0522

0.6

1.0000

0.9613

0.8426

0.7530

0.6415

0.5110

0.3585

0.1885

0.0965

0

0.7-0.9

1.0000

0.9600

0.8400

0.7500

0.6400

0.5100

0.3600

0.1900

0.0975

0

..".

c:n

f-'

452

Basic Ship Propulsion TableA3.5 Coefficients C T for K T Polynomial

I <,

-; ~o.

\.

1 2 3 4 5 6 7 8 9 10 11 12 13 , 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

29 30 31 32

i

0 1 0 0 2 1 0 0 2 0 1 0 1 0 0 2 3 0 2 3 1 2 1 1 3 0 1 0 0 1 2 3

J

0 0 1 2 0 1 2 0 0 1 1 0 0 3 6 6 0 0 0 0 6 6 6 3 3 3 0 2 0 0 0 0

k D

0 0 0 1 1

1 0 0 0 0 1 1 0 0 0 1 2 2 2 2 2 2 0 0 1 2 2 0 0 0 0

I 0 0 0 0 0 0 0 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1

1 1 1 2 2 2 2

CT(i,j,k,l)

+0.00880496 -0.204554 +0.166351 +0.158114 -0.147581 -0.481497 +0.415437 +0.0144043 -0.0530054 +0.0143481 +0.0606826 -0.0125894 +0.0109689 -0.133698 +0.00638407 -0.00132718 +0.168496 -0.0507214 +0.0854559 -0.0504475 +0.010465 -0.00648272 +0.010465 +0.168424 -0.00102296 -0.0317791 +0.018604 -0.00410798 -0.000606848 -0.0049819 \ +0.0025983 -0.000560528

.~ ':;

., ~"l

<: , r~ ;;~

Ij

.~

~!~

1

~

~

~

~i

~

~,

~

~ ~

I~

",1

g i

;,

«

J ~

i

N ii

I

I

j


454 Table A3.6 (Contd.)

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

2 0 0 3 3 0 3 0 1 0 2 0 1 3 3 1 2 0 0 0

2 2 3 2 6 2 00 3 0 6 0 0 1 6 1 0 2 2 2 3 2 6 2 1 0 2 0 6 0 0 .1 0 1 2 1 6 1 0 2

0 0 0 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2

0 3

3 3

2 2

2 2

0 1

6 6

2 2

2 2

+0.0417122 -0.0397722 -0.00350024 -0.0106-854 +0.00110903 -0.000313912 +0.0035985 -0.00142121 -0.00383637 +0.0126803 -0.00318278 +0.00334268 -0.00183491 +0.000112451 -0.0000297228 +0.000269551 +0.00083265 +0.00155334 +0.000302683 -0.0001843 -0.000425399 +0.0000869243 -0.0004659 +0.0000554194

(b) Gawn Series (Admiralty Experimental Works 20 inch Series)

Geometry

The geometrical particulars of the Gawn Series are given in Ta

ble 4.1 and Fig. 4.4. The developed blade outline is an ellipse. The

centre of the ellipse is at a distance of O.275D from the centre of

the propeller and the semi-axes of the ellipse, a and b, are given

by:

I G

\ ~

;

J

455

Appendix 3

a =

b D

O.225D

(A3.5)

where the coefficients c(i,j) are given in Table A3.7. TableA3.7

Blade Outline

c(i, j)

i

J

0

0

O.53836624e-02

1

0

O.37401561e+OO

2

0

-O.34196916e-02

0

1

-O.58167525e-02

1

1

O.25190058e-Ol

2

1

-O.90211731e-02

0

2

O.14566425e-02

1

2

-O.67853528e-02

2

2

O.35134575e-02

KT, KQ Polynomials

KT =

~CT(i,j,kll) Ji (~)j (1~)k t,J,k

KQ =

~ Cdi,j, k,l) Ji (~) j (1~)

(A3.6) k

t,J,k

The coefficients CT and CQ are given in Table A3.8.

;

l


456

\.

TableA3.8

tip

CT and C Q in the Polynomials for K T and lOKQ

\

t

J

k

0

0

0

0.97408086e - 01

0.63692808e - 01

1

0

0

-0.21572784e + 00

-0.8008.9413e - 01

2

0

0

-0.20117083e + 00

-0.16184720e + 00

3

0

0

0.6885705ge - 01

-0.21572175e - 01

0

1

0

0.1609927ge + 00

0.77811621e - 01

1

1

0

0.29145467e + 00

0.2470281ge + 00

2

1

0

0.19188970e - 01

0.62403660e - 01

3

1

0

-0.5595656ge - 01

-0.53570643e - 01

0

2

0

-0.13744167e -01

0.14384213e + 00

1

2

0

-0.10454782e + 00

-0.13415100e + 00

2

2

0

0.38724232e - 01

0.35848316e - 01

3

2

0

0.61652786e - 02

0.12422566e - 01

0

0

1

-0.40506446e+ 00

-0.48559698e + 00

1

0

1

0.19004023e + 00

0.74163526e + 00

2

0

1

-0.13912685e + 00

-0.72897837e - 01

3

0

1

0.66181317e - 01

0.10767796e + 00

0

1

1

0.10065955e + 01

0.84972864e + 00

1

1

1

-0.11633307e + 01

-0.21723540e + 01

2

1

1

0.3212153ge + 00

-0.87905906e - 01

3

1

1

-0.12919416e - 01

0.21879166e + 00

0

2

1

-0.23608254e + 00

0.30810708e + 00

1

2

1

0.75812680e + 00

2

2

1

o.50920802e + 00 -0.20886181e + 00

-0.19750497e + 00

3

2

1

0.16416967e - 01

-0.49353302e - 01

CT(i,j, k)

~1

,~

'1

CQ(i,j, k)

.,

,,

i

~

'.,

f' ':1

~

~ "::'

tr';

More elaborate regression coefficients have been given by Blount and Hubble (1981).

~' ~ ~

457

Appendix 3

(c) Ka Series in Nozzle 19A (MARIN)

Geometry The nomenclature used is the same as that used for the B Series. Table A3.9 Blade Outline and Blade Thickness

~

r/R

Cl

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1.3222 1.5081 1.6774 1.8314 1.9690 2.0844 2.1675 2.2183 2.4195

xo/c 0.5501 0.5277 0.5134 0.5055 0.5013 0.5000 0.5000 0.5000 I 0.5000'

xm/c

tiD

0.3498 0.3976 0.4602 I 0·51913 0;.4998 0.5000 0.5000 0.5000 0.5000

0.0400 0.0352 0.0300 0.0245 0.0190 0.0138 0.0092 0.0061 0.0050

f

TableA3.10

,

Blade Section Face and Back Ordinates

f

I

I

(a)

x/x m 0

0.05

0.10

r/R

0.20

0.40

0.60

0.80

1.00

Values ofu

0.2

0.3333

0.2062

0.1604

0.1052

0.0437

0.0146

0.0021

0

fi'

0.3

0.2118

0.1030

0.0828

0.0615

0.0272

0.0083

0.0012

0

f

0.4

0.1347

0.0444

0.0389

0.0292

0.0139

0.0042

0

0

r

0.5

0.0781

0.0153

0.0136

0.0102

0.0051

0.0017

0

0

0

0

0

0

0

0

0

I

I

r

I.

1-

0.6-1.0 0

I 458


~a ;~~

:1

TableA3.10

l,i

~i

(b)

!:~

(x - xm)/(c x m ) 0

0.20' 0.40

r/R

0.60

0.80

1.00

Values of u

0.2

0

0

0.0010

0.0177

0.0729

0.2021

0.3

0

0

0

0.0107

0.0462

0.1385

0.4

0

0

0

0.0056

0.0236

0.0917

0.5

0

0

0

0.0017

0.0068

0.0662

0.6-1.0

0

0

0

0

0

0

0.60

0.80

1.00

'.t

TableA3.10

(c)

x/x m 0

0.05

0.10

r/R

0.20

0.40

Values of v

0.2

0.3333

0.4802

0.5479

0.6552

0.8156

0.9229

0.9813

1.0000

0.3

0.2118

0.3787

0.4615

0.5917

0.7834

0.9089

0.9775

1.0000

0.4 \0.1347

0.3027

0.3861

0.5292

0.7500

0.8931

0.9722

1.0000

0.5

0:0781

0.2377

0.3158

0.4686

0.7097

0.8727

0.9677

1.0000

0.6

0

0.2044

0.2859

0.4358

0.6826

0.8589

0.9647

1.0000

0.7 0

0.2288

0.3079

0.4531

0.6924

0.8633

0.9658

1.0000

0.8

0

0.2690

0.3439

0.4816

0.7084

0.8704

0.9676

1.0000

0.9

O.

0.3187

0.3887

0.5175

0.7294

0.8809

0.9717

1.0000

1.0 0

0.3231

0.3925

0.5200

0.7300

0.8800

0.9700

1.0000

Appendix 3

459

TableA3.10

(d)

o

0.20

0.40

r/R

0.60

0.80

1.00

Values of v

0.2 0.3

1.0000 1.0000

0.9500 0.9586

0.8250 0.8414

0.6542 0.6770

0.4552 0.4367

0.2021 0.1385

0,4

1.0000

0.9625

0.8569

0.6750

0.4292

0.0917

0.5

1.0000

0.9660

0.8642

0.6876

0.4245

0.0662

0.6

1.0000

0.9647

0.8589

0.6826

0.4358

0

0.7

1.0000

0.9658

0.8633

0.6924

0.4531

0

0.8

1.0000

0.9676

0.8704

0.7084

0.4816

0

0.9

1.0000

0.9717

0.8803

0.7294

0.5175

0

1.0

1.0000

0.9700

0.8800

0.7300

0.5200

0

Geometry of Nozzle 19A: Komenclature D : Propeller diameter

f

e

I

Length of nozzle

n r0

f

I

Ra.dius of inner surface of nozzle :

Radius of outer surface of nozzle

x : Distance along axis from leading edge of nozzle

t

Yi : Offset of inner surface of nozzle

r

~.

I

Clearance between propeller and nozzle

Yo : Offset of outer surface of nozzle I

D = 0.5

l ---

ri = 0.5D

+ e + Yi

ro

=::

0.5D+e+yo

(A3.7)

460

Basic Ship Propulsion RADIUS TRAILING LEADING EDGE EDGE .0.01890 I 0.02875 I

r.I

0.50

J<-----. X

----'71'"

Figure A3.2: Geometl:V of Nozzle 19A.

TableA3.11

Nozzle 19A Profile

x/I

o 0.0125 0.0250 0.0500 0.0750 0.1000 0.1500 0.2000

yo/I

0.1825 0.1466 0.1280 0.1007 0.0800 0.0634 0.0387 0.0217

0.1825 0.2072 0.2107 0.2080 0.1969 0.1858 0.1747 0.1636

461

Appendi.\: 3 Table A3.11 (Contd.)

0.0110 0.0048 0 0 0 0.0029 0.0082 0.0145 0.0186 0.0236

0.2500 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 0.9500 1.0000

i

0.1525 0.1414 0.1302 0.1191 0.1080 0.0969 0.0858 0.0747 0.0692 0.0636

I, \

r

,f

KT' KQ Polynomials

, (PY D J3.

r

,[!,

Total thrust coefficient:

KT

- ~ CT(i,j)

(A3.8)

t,J

Torque coefficient

=

KQ

~ CQ(i,j) (~y Jj

(A3.9)

t,J

Duet thrust coefficient :

KTD

= ~ CTD(i,j) (~) i

Jj

t,J

TableA3.12 Coefficients CT, C Q and CTD in Polynomials for K T , KQ and KTD (a) Propeller: Ka 3.65 in Nozzle 19A

fi

(

r.

L

i

J

CT(i,j)

CQ(i,j)

CTD(i,j)

o o o o

0 1 2 3

0.028100 -0.143910 0.0 -0.383783

0.006260 0.0 -0.017942 0.0

0.154000 0.115560 -0.123761 0.0

(A3.10)

~

462

B8Sic Ship Propulsion Table A3.12(a) (Contd.)

0 0

a 1 1 1 1 2 2 3 3 3 4 5 6

6 6

4, 5 6

a 1 2 6 0 2 0 2 6 3 1 0 1 2

-0.008089 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.429709 0.0 -0.016644 0.0 0.0 0.0 0.0 0.671268 0.286926 0.0 -0.182294 0.040041 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 . -0.003460 0.0 -0.017378 -0.000674 0.001721 0.0

0.0 -0.741240 0.646894 -0.542674 0.749643 0.0 -0.162202 0.972388 1.468570 -0.317664 -1.084980 -0.032298 0.199637 0.060168 0.0

0.0

0.0

TableA3.12 (b) Propeller: Ka 4.55 in Nozzle 19A

z

j

0 0 0 1 0 2 0 3 0 4 0 5 0 6 0 7 1 0 1 1 1 2 1 3

CT(i,j)

CQ(i,j)

CTD(i,j)

-0.375000 -0.203050 0.830306 -2.746930 0.0 0.0 0.067548 0.0 2.030070 -0.392301 -0.611743 4.319840

-0.034700 0.018568 0.0 0.0 -0.195582 0.317452 -0.093739 0.022850 0.158951 -0.048433 0.0 0.024157

-0.045100 0.0 0.0 -0.663741 -0.244626 0.0 0.0 0.0 0.244461 -0.578464 1.116820 0.751953

I.. ' I,

~

~,

463

Appendi..x 3 Table A3.12(b) (Contd.)

0.0 0.0 4 -0.341290 -0.123376 0.0 0.0 5 0.0 0.0 -0.089165 6 0.0 0 -3.031670 -0.212253 -0.146178 0.0 0.0 1 0.0 0.0 -0.917516 2 -2.007860 3 0.0 0.0 2.836970 0.156133 0.068186 0 1 0.0 0.174041 0.0 2 0.0 0.0 0.102331 0.391304 0.0 0.0 3 0.0 0.0 0 -0.994962 0.0 0.030740 0.0 1 0.0 0.073587 0.0 2 -0.031826 0.0

0.0 0 0.0

0.015742 -0.014568 1 I 2 -0.109363 0.0 0.0 0.0 0.0

4 0.043862 6 0 0.043782 0.007947 -0.008581

0.0 0.0 6 2 0.038275 -0.021971 0.0 6 4 0.0 6 6 0.0 0.000700 0.0

1 1 1 2 2 2 2 3 3 3 3 4 4 4 5 5 5 5

TableA3.12

(c)

Propeller: Ka 4.70 in Nozzle 19A

i',

(

i I:

l

t

j

CT(i,j)

CQ(i, j)

CTD(i,j)

0 0 0 0 0 0 0

0 1

0.030550 -0.148687 0.0 ·-0.391137 0.0 0.0 0.0

0.006735 0.0 -0.016306 0.0 -0.007244 0.0 0.0

0.076594 0.075223 -0.061881 -0.138094 0.0 -0.370620 0.323447

2

3 4 5 6

"

I Basic Ship Propulsion

464

ti

Table A3.12(c) (Contd.) 1

0

1 1

1 2

1 2 2

6

a 2

3 a 3 2 3 6 4 0 4 3 5 1 6 0 6 1 6 2

0.0 -0.432612 0.0 0.0 0.667657 0.285076 -0.172529 0.0 0.0 0.0 0.0 0.0 0.0 -0.017293 0.0

0.0 0.0 -0.024012 0.0 0.0 0.005193 0.046605 0.0 0.0 -0.007366 0.0 0.0 -0.001730 -0.000337 0.000861

,

-0.271337 -0.687921 0.225189 -0.081101 0.666028 0.734285 -0.202467 -0.542490 -0.016149 0.0 0.099819 0.030084 0.0 0.0 -0.001876

a~ ~

>!i ~,

~~ .~

.i

~

•

'" ~,

f

ti

~)\

~

'il

~

1 J I

TableA3.12

(d)

Propeller: Ka 5.75 in Nozzle 19A J

0 a 0 1 0 2 0 3 0 4 1 1 I 2 2 0 2 2 3 0 4 0 6 1 6 2

CT(i,j)

CQ(i,j)

CTD(i,j)

0.033000 -0.153463 0.0 -0.398491 0.0 -0.435515 0.0 0.664045 0.283225 -0.162764 0.0 -0.017208 0.0

0.007210 0.0 -0.014670 0.0 -0.006398 0.0 -0.031380 0.0 0.010386 0.053169 -0.014731 0.0 0.0

-0.000813 0.034885 0.0 -0.276187 0.0 -0.626198 0.450379 0.359718 0.0 -0.087289 0.0

0.0

-0.003751

ApPENDIX

4

Propulsion Factors Nomenclature

J,1.E

Expanded blade area ratio of/the propeller I

Ao

Disc area of the propeller

B

Breadth of the ship

CA

Correlation allowance

CB

Block coefficient

CF

Frictional resistance coefficient

CM

Midship section coefficient

Cp

Prismatic coefficient

Cpv

Vertical prismatic coefficient

Cv

Viscous resistance coefficient

Cw

Waterplane coefficient

D

Propeller diameter

Fn

Froude number

h

Depth of immersion of the propeller axis

465

J.

_


466 L

Length of the ship

LCB

Longitudinal coordinate of the centre of buoyancy of the ship forward of amidships as a percentage of the length of the ship

P

Pitch of the propeller

S

\-Vetted surface of the ship

t

Thrust deduction fraction

T

Draught of the ship

T.4.

Draught at after perpendicular

TC

Propeller tip clearance from hull

"! -:

1 -,1

1 ~ -"

4 ~ a .~

'j

V'

Ship speed

W

Wake fraction

WF

Froude wake fraction

~

~

,\ J -~

~1

1

.',

e

Rake angle

7]R


,

1/J

Angle of bossing to the horizontal

i

y

Displacement volume of the ship

l

\Vake Fraction Taylor (1933)

Single screw ships

w

=

1.7489

c1 -

1.8612 C s

+ 0.7272

(A4.1)

467

Appendix 4 Twin screw ships w

=

1.7643

c1- 1.4745 CB + 0.2574

(A4.2)

Schoenhetr (Rossell and Chapman, 1939)

Single screw ships

w _ 0.10+4.5

CpvCpB/L

(7 - 6 Cpv)( 2.8 - 1.8 Cp )

+ 0.5 ( T-h ----;y- k'

D ) B - k' e

=

0.3 for normal sterns

=

0.5-0.6 sterns with cutaway deadwood

(A4.3)

e in radians. Twin screw ships With bossings and outward turning propellers W

=

2 C~ (1- CB) + 0.2 cos ~1/J 2

-

(A4.4)

0.02

With bossings and inward turning propellers W

=

2C~ (1- CB) + 0.2cos ~(90 2

-1/J) -

0.02

(A4.5)

With propellers supported by struts W

(A4.6)

= 2 C~ (1- CB) + 0.04

Burrill (1943) Single screw ships W

WF

=- I- w

0.285 - 0.417 CB + 0.796

c1

(A4.7)

\_----------~------

468


Twin screw ships With bossings at 10 degrees to the horizontal Ii:

WF

= - - = 0.171 - 0.847 CB + 1.341 1-

lJ.1

C1

(A4.8)

With bossings at 30 degrees to the horizontal

tc

WF = - - =

1 - u:

0.052 - 0.648 CB + 1.138 C~

(A4.9)

Harvald (1983) Harvald's data are given in the form of diagrams, which have been converted to the following formulas: Single screw ships I (A4.10)

where the + sigr: in the second term applies to U-form hulls and the - sign to V-form hulls and the constants a(i,j), b(i) and c(i) are given in Table A4.1. TableA4.1

\\-ake Coefficients for Single Screw Ships

z .j

o

0

1

0

2

0

3

0

o

1

1

1

2 3

1 1

aU, j)

i

-0.25561270e+Ol 0.15080732e+02 -0.276E0372e+02 0.16433S67e+02 0.17220..,l05e+02 -0.930-lS350e+02 0.165.57534e+03 -O.9206·568ge+02

0 1 2 3 4 5

b(i)

c(i)

-0.13033033e+02 0.59075583e+OO 0.1l350812e+03 -0.33099666e+02 -0.38820110e+03 0.70200373e+ 03 0.6507232ge+03 -0.7259522ge+04 -0.53369306e+03 0.29089242e+05 0.17128503e+03

Appendix 4

469

Twin screw ships

w

=

~a(i,j)Ck (~)j +ow

(A4.11)

'" where Ow is 0.04 for ships with pronounced V-sections and -0.0i\ for ships with shaft brackets instead of bossings. The coefficients a( i, j) are as given in Table A4.2: TableA4.2

Wake Coefficients for Twin Screw Ships

a(i/j)

j

o 1 2

3

o 1 2 3

o o o

-0.20940003e+01 0.10263335e+02 -0.16825002e+02 0.94166677e+01 0:18200002e+02 -0.8750000ge+02 0.14250001e+03 -0.75000007e+02

o 1

1 1 1

B.S.R.A. (Parker, 1966)

Single screw ships

(A4.12) where

l------'

ao

=

a4

= 0.2525

-0.8715

al

=

2.490

as = 0.226

a2 a6

=

-1.475

=

-7.176 x 10- 3

a3 =

-0.3722

470

Basic Ship PropulBion SLCB

=

LCB-20(CB -0.675)

V in knots, L in feet.

Holtrop (1984) Single screw ships

=

W

Cg C20

c,,~

(0.050776 + 0.93405 Cll

+ 0.27915 C20

1

(L(l!C PI

~~PI)

))

+ C1SC20

(A4.13)

Cg

-

B SI( LDTA) when B ITA :5 5

Cg

-

S(7BITA -25)/(LD(BITA -3)) when BITA ~ 5

Cg

=

cs when

Cll

=

1'.41 D

Cll=

Cg

:5 28 and

Cg

=

32 - 16/( Cg

-

24) when

Cg ~

28

when TAl D :5 2

0.833333 (TAID)3

+ 1.33333 when TAID

~ 2

~

C19

= 0.12997/(.0.95 - CB ) - 0.11056/( 0.95 - Cp) when Cp

CIS

= 0.185671 (1.3571 - CM) - 0.71276 + 0.38648 Cp when Cp

C20

=

1 + 0.015 Cstern where given in TableA4.3.

Cstern

0.7 ~

0.7

depends upon the type of stern as

I ,

r

Ii ~

~

t f

.J

471

Appendix 4

Wake Coefficients for Different Stern Types

Stern Type

Cstern

Pram with gondola V shape sections Normal stern U shape sections with Hogner stern

CPl

=

-25.0 -10.0 0.0 10.0

1.45 Cp - 0.315 - 0.0225 LCB

Single screw ships with open stern w = 0.3 CB

+ 10 Cv CB -

(A4.14)

0.1

Twin screw ships

+ 10Cv CB -

w = 0.3095 CB

0.23 DjvBT

(A4.15)

Papmel (Reference not known) w = 0.165C~

"\1 1 / 3 )

(

D

~ +F

n -

0.2

(A4.16)

where k = 1.0 for single screw ships and k = 2.0 for twin screw ships.

Thrust Deduction Fraction Schoenherr (Rossell and Chapman, 1939) Single screw ships

t = kw

l

(A4.17)

-----------------


472 where

k -

0.50-0.70 with streamlined or contra rudders;

k -

0.70-0.90 with double plate rudders attached to square rudder posts;

k -

0.90-1.05 with single plate rudders.

Twin screw ships (A4.18) (A4.19)

Edstrand (Reference not known) Single screw ships

t CB - == 1.57 - 2.3 Cw w

+ 1.5 C B

(A4.20)

+ 1.5 CB

(A4.21)

Twin screw ships

t CB

til = 1.67 - 2.3 Cw

Harvald (1983) Single screw ships (A4.22)

where the + sign is for U forms, the - sign is for V forms and the coefficients a( i. j) are given in Table A4.4.

473

Appendix 4 TableA4.4

Coefficients for Thrust Deduction (Single Screw Ships)

a(i, j)

t

j

0

0

0.44197757e+00

1

0

-0.13492692e+Ol

2

0

0.1l528267e+Ol

0

1

-0.1l373258e+Ol

1

1

0.60955480e+Ol

2

1

-0.4J!455395e+O,1

Twin screw ships

t -

~ a(i) ck + 4 (~ ,

0.03)

-6 (Tf -0.005) +6.t

(A4.23)

where b.t = 0 with bossings, b.t = -0.02 with struts and the coeffi cients a(i) are as given in TableA4.5. TableA4.5

Coefficients for Thrust Deduction (Twin Screw Ships) "

I'"

I;

~"

I'

~

V I

i

,; ,

~ "

"

l

i

a(i)

o

o.17300000e+00

1

-0.40928571e+OO

2

0.64285714e+OO


474

B.S.R.A. (Parker, 1966) Single screw ships

.

+ bs (GBVVL) 3 + b6::r + b7 D t + be 8LGB + bg GB 8LGB bo = -0.1158

b1

b4 = 0.05432

bs = -0.02419

be = 0.05171

bg = -0.08622

DB Dt = -

\7~

= 0.08859

b2

= 0.3133

b3

b6 = -0.4542

(A4.24)

= 0.2758 b7 = 0.6044

8LGB = LGB - 20 (GB - 0.675)


Holtrop (1984)

" screw ships Single , t =

0.25014 (B j L )0.289S6 ( VBT j D )0.2624 ( 1 _ Gp + 0.0225 LGB )0.01762

+ 0.0015 Gstern

(A4.25)

Single screw ships with open sterns

t = 0.10

(A4.26)

Twin screw ships t = 0.325GB - 0.1885 DjvBT

(A4.27)

475

Appendix 4 Relative Rotative Efficiency Schoenherr (Rossell and Chapman, 1939) Single screw ships TJR = 1.02

(A4.28)

average

Twin screw ships

TJR = 0.985

(A4.29)

average

van Manen (Reference not known) Single screw ships TJR

I

l ~'.

I

=

1.02-1.07

(A4.30)

average = 1.05

Twin screw ships TJR

=

0.95-1.00

average

=

(A4.31)

0.97

B.S.R.A. (Parker, 1966)

Single screw ships

t

(A4.32)

I

~

!

'f

I ~.

co = 1.716

(

Ca = -0.0308


~------

Cl

= -2.378 C4

= 0.6931

C2

= 1.742

476


Holtrop (1984)

Single screw ships

AE + 0.07424 (Gp -

TlR = 0.9922 - 0.05908 Ao

0.0225LCB)

(A4.33)

Single screw ships with open sterns TlR = 0.98

(A4.34)

Twin screw ships TlR

=

0.9737 + 0.111 (Gp ~ 0.0225 LGB) - O.06325P/D

(A4.35)

ApPENDIX

5

Propeller Blade Section Pressure Distribution ! TableA5.1

I

Effect of Camber

NACA Mean Lines

a= 0.8

= 1.0 x/c

CLi

\

I ;

~

~

~.' ~ E ~

l

0.0125 0.0250 0.0500 0.0750 0.1000 0.1500 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000

Qi

!

= 0.8 (mQdified) CLi = 1.0 Qi = 1.40 a

= 1.54

0

a 0

eLi

= 1.0

= 1.0

(Xi

=0

0

·6. Vf/ Vo

x/c

6.Vf/Vo

x/c

6.Vf/VO

0.278 0.278 0.278 0.278 0.278 0.278 0.278 0.278 0.278 0.278 0.278 0.278 0.278

0.0125 0.0250 0.0500 0.0750 0.1000 0.1500 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000

0.273 0.273 0.273 0.273 0.273 0.274 0.274 0.274 0.275 0..276 0.276 0.277 0.278

0.0125 0.0250 0.0500 0.0750 0.1000 0.1500 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000

0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250

477

-------~

..



0.9000 0.9500 1.0000

0.139 0.069

o

0.9000 0.9500 1:0000

0.147 0.092 0

0.9000 0.9500 1.0000

0.250 0.250 0.250

Table A5.2 Effect of Thickness Distribution (a)

Thickness Distribution: NACA 16 Values of t::.. Vi/VO xlc

tic = 0.06

0 0 0.0125 0.029 0.0250 0.042 0.0500 0.047 0.0750 0.051 0.1000 0.053 0.1500 0.055 0.2000 0.057 0..3000 0.060 0.4000 0.064 0.066 0.5000 0.6000 0.068 0.7000 0.064 0.8000 0.057 0.9000 0.017 0.9500 -0.019 1.0000 0

tic = 0.09

tlc,=

0 0.021 0.053 0.067 0.073 0.076 0.081 0.085 0.091 0.096 0.100 0.106 0:099 0.075 0.022 -0.031 0

0 '0'.001 0.053 0.083 0.094 0.099 0.106 0.112 0.121 0.128 0.134 0.137 0.129 '0.097 0.025 -0.047 0

0.12,

.tlc = 0.15

tic = 0.18

tic = 0.21

0 -0.022 ·0.051 0.095 0.113 0.121 0.130 0.139 0.152 0.161 0.168 0.172 0.161 0.120 . 0.026 -0.065 0

0 -0.050 0.045 0.103 0.128 ' 0.141 0..154 0.165 0.183 . 0.194 0.203 ' 0.205 0.192 0.143

0 -0.091 0.031

0~025

-0.085 O'

0~105

0.138 0.159 0.179 0.191 0.214 0.227 0.239 0.237 0.223 0.166 0.019 -0.105 0

Appendix 5

479 TableA5.2 Effect of Thickness Distribution (b)

Thickness Distribution: NACA 66 Values of 6 Vi/Vo xlc

r tt

f ~

t;

~

:,

r f f

t

,

I I f

r t

tic = 0.06

0 0 0.0125 0.031 0.0250 0.035 0;0500 0.042 0.0750 0.048 0.1000 0.052 0.1500 0.058 0.2000 0.062 0.3000 0.067 0.4000 0.070 0.5000 0.073 0.6000 0.075 0.7000 0.057 0.8000 0.020 0.9000 -0.026 0.9500 -0.057 1.0000 -0.093

= 0.12

tic = 0.09

tic

0 0.018 0.039 0.058 0.069 0.077 0.085 0.091 0.100 0.105 0.110 0.114 0.083 0.025 -0.043 -0.084 -0.136

0 -O.OlD 0.036 0.067 0.085 0.097 0.112 0.122 0.134 0.142 0.148 0.154 0.105 0.026 -0.062 -0.112 -0.171

tic = 0.15

tic = 0.18

tic = 0.21

0 -0.036 0.027 0.078 0.099 0.114 0.134 0.148 0.164 0.175 ; 0.184 0.192 0.122 0.026 -0.080 -0.137 -0.201

0 -0.103 0.002 0.074 0.111 0.134 0.162 0.180 0.202 0.217 0.228 0.238 0.141 0.022 -0.104 -0.168 -0.234

0 -0.131 -0.024 0.069 0.116 0.148 0.185 0.208 0.236 0.255 0.269 0.284 0.155 0.015 -0.127 -0.195 -0.266

I

TableA5.3 Effect of Angle of Attack

(a) Thickness Distribution: NACA 16 Values of 6 Vai/Vo ,.

i

~;

xlc

tic = 0.06

tic = 0.09

tic = 0.12

tic = 0.15

tic = 0.18

0 0.0125

5.471 1.376

3.644 1.330

2.624 1.268

2.041 1.209

1.744 1.140

I, ;~

r..,6IiW

d

tic

= 0.21

1.574 1.069


480 TableA5.3(a) (Contd.)

0.0250 0.0500 0.0750 0.1000 0.1500 0.2000 0.3000 0.4000 0.5000 0.6000 0.7000 0.8000 0.9000 0.9500 1.0000

0.980 0.689 0.557 0.476 0.379 0.319 0.24-1 0.196 0.160 0.130 0.104 0.077 0.049 0.032 0

0.964 0.684 0.554 0.475 0.378 0.319 0.245 0.197 0.160 0.131 ' 0.103 0.076 0.047 0.030 0

0.942 0.677 0.551 0.473 0.378 0.319 0.245 0.197 0.161 0.131 0.102 0.075 ,0.045 0.027 ,0

I ~

0.916 0.668 0.547 0.471 0.377 0.318 0.245 0.197 0.161 0.131 0.102 0.074 0.043 0.025 0

0.883 0.657 0.541 0.4.68 0.376 0.318 0.245 0.198 0.162 0.131 0.102 0.073 0.042 0.024 0

0.828 0.640 0.534 0.463 0.374 0.317 0.245 0.198 0.162 0.131 0.102 0.072 0.041 0.023 0

TableA5.3 Effect of Angle of Attack (b) Thickness Distribution: NACA 66

Values of l::,.Vai/Vo xjc

tjc = 0.06

tjc = 0.09

tjc = 0.12

tjc = 0.15

0 0.0125 0.0250 0.0500 0.0750 0.1000 0.1500 0.2000 0.3000 0.4000 0.5000

4.941 1.500 0.967 0.695 0.554 0.474 0.379 0.320 0.245 0.197 0.161

3.352 1.340 0.940 0.686 0.552 0.473 0.379 0.322 0.246 0.197 0.161

2.569 1.237 0.913 0.674 0.549 0.473 0.380 0.323 0.246 0.197 0.162

2.139 1.172 0.895 0.663 0.547 0.473 0.381 0.323 0.248 0.200 0.163

tjc

= 0.18

1.773 1.121 0.858 0.649 0.545 0.472 0.381 0.323 0.250 0.201 0.163

tjc = 0.21

1.547 1.054 0.828 0.635 0.542 0,472 0.381 0.324 0.251 0.202 0.165

J

Appendix 5.

481 Table A5.3(b) (Contd.)

0.6000 0.7000 0.8000 0.9000 0.9500 1.0000

0.130 0.102 0.075 0.047 0.030

o

0.130 0.132 0.131 0.100 0.098 0.096 0.071 0.069 0.065 0.043 0.040 0.039 0.028 0.031 0.025 000

0.131 0.095 0.061 0.037 0.022

0.132 0.093 0.058 0.034 0.020

o

o

i.

I

, i! ,~:

,

I

l "

.;" .

_

ApPENDIX

6

(joldstein Factors K.

= a

+b

(:r ) + (:r ) + c

2

TableA6.1

Number of Blades, Z

d

(:J

(A6.1)

3

= 3

r/R

a

b

c

d

0.200 0.30b 00400 0.500 0.600 0.700 0.800 0.850 0.900 0.950 0.975

0.1788690ge+01 0.12344242e+01 0.8588666ge+00 0.62018788e+00 OA1573334e+00 0.27236363e+00 0.14103636e+00 0.10892727e+00 0.26095757e+00 0.34436364e-01 0.14018182e-01

-0.31486830e+00 -0.13838772e+00 0.22855088e-0l 0.12671252e+00 0.21640132e+00 0.25907344e+00 0.28452680e+00 0.25856760e+00 0.11420357e-01 0.17165734e+00 0.12805361e+00

OA0256411e-01 0.24405595e-01 0.12983683e-02 -0.13955711e-01 -0.28552448e-01 -0.34629371e-01 -0.40822845e-0l -0.35081584e-01 0.41284382e-0l -0.21487178e-01 -0.16160838e-0l

-0. 16456876e-02 -0.14421134e-02 -0.32012432e-03 OA5532247e-03 0.13177933e-02 0.16923076e-02 0.23263404e-02 0.1953379ge-02 -0.54623154e-02 0.11934732e-02 0.95571094e-03

TableA6.2

Number of Blades, Z r/R

a

0.200 0.300 0.400

0.1749636 le+0 1 0.13368727e+01 0.10046545e+01

b f

= 4

c

-0.33573776e+00 -0.19570281e+00 -0.2990442ge-01

482

0.50925408e-01 0.36153845e-01 0.7652680ge-02

d -0.26573427e-02 -0.22237762e-02 -0.53613057e-03

Appendix 6


0.500 0,600 0,700 0.800 0.850 0.900 0.950 0.975

0.74438787e+OO 0,5082787ge+00 0.36886668e+00 0.21533333e+00 0.16254544e+00 0.9437575ge-01 0.26442423e-01 0.30242424e-02

0.10496193e+00 0.23900738e+00 0.27456838e+00 0.30709362e+00 0.28706294e+00 0.26828283e+00 0.22522765e+00 0.16694678e+00

-0,i5128205e-0l -OA1976690e-01 -OA3508161e-0l -OA8083916e-0l -OA1976690e-01 -0.3878787ge-01 -0.33561774e-0l -0.23643358e-01

0.71173272e-03 0.25905205e-02 0.24397825e-02 0.28236208e-02 0.23682984e-02 0.22626263e-02 0.20916860e-02 0.14390054e-02

TableA6.3

Number of 13lades, Z = 5

r/R

a

b

0.200 0.300 00400 0.500 0.600 0.700 0.800 0.850 0.900 0.950 0.975

0.16373152e+01 0.12795818e+01 0.10774000e+01 0.85581213e+00 0.65652120e+00 OA611091Oe+OO 0.3235999ge+00 0.22852121e+00 0.16675758e+00 0.5747878ge-01 0.15157576e-0l

-0.30654624e+00 -0.15718766e+00 -0.63241085e-01 0.63054390e-01 0.17247047e+00 0.26796621e+00 0.28312004e+00 0.29787374e+00 0.26352641e+00 0.24161461e+00 0.18922222e+00

c

0.5P608393e-0l 0.2779020ge-01 0.13892774e-01 -0.97389277e-02 -0.29790210e-01 ....iOA7247086e-01 ':"OA4466201e-01 . -0,46969697e-01 -0.37958041e-01 -0.3760l400e-01 -0.28424243e-01

'l'ableA6.4

Number of Blades, Z

I'

I

I

~.

f

!

l

r/R

a

b

0.200 0.300 00400 0.500 0.600 0.700 0.800 0.850 0.900 0.950 0.975

0.14782788e+01 0.1315187ge+01 0.11262424e+0l 0.9269757ge+00 0.73245454e+00 0.57758790e+00 0.38398787e+00 0.31183636e+00 0.20169698e+00 O.19169698e-01 0.40769696e-01

-0.21982946e+00 -0.18757537e+00 -0.94499610e-0l O.32015931e-01 0.1542062ge+00 0.22007382e+00 0.28972650e+00 0.2791526ge+00 0.28262860e+00 O.3273721ge+00 0.1912940ge+OO

d -0.29106450e-02 -0.16177156e-02 -0.97902096e-03 004 7241646e-03 0.16985236e-02 0.2871794ge-02 0.25128205e-02 0.28080808e-02 0.21740482e-02 . 0.24413364e-02 0.17777778e-02

= 6

c

d

0.33473194e-01 -0.16891997e-02 0.35438228e-01 -0.21507381e-02 0.21074591e-01 -0.14840715e-02 0.24397824e-03 -0.50023310e-02 0.20186480e-02 -0.30512821e-01 0.23294482e-02 -0.39191142e-01 0.30287490e-02 -0.49876455e-0l 0.24568764e-02 -O.4341258ge-01 -OA3582752e-01 , 0.25905205e-02 0.51686093e-02 -0.64843826e-01 0.17700078e-02 -0.28200466e-01

f··,

ApPENDIX

7

Cavitation Buckets The following formulas are based on material given in Breslin and Andersen (1994). The maximum and minimum values of angle of attack within which there is no cavitation are given by:

t + ! \1[- C

Qmax

=

a1

Qmin

=

al -,. -

c

f

c

.

a2

c

t

a2 -

c

p min

t]

+ a3. !c + a4 ·c

[ Cp min + a3 !c

t]

(A7.1)

- a4 c .

where the constants aI, a2, a3, a4 depend on the mean line and the thickness distribution of the blade section and are given in TableA7.1. TableA7.1

: Coefficients of Minimum and Maximum Angles of Attack

Mean Line a = 0.8 a = 0.8 (modified) a = 0.8 (modified) a = 1.0 a = 1.0

Thickness Distribution

:\AOA-16 :\AOA-66 (modified) :\AOA-16 ~AOA-16

:\AOA-66 (modified)

484

al

22.6804 21.0495 22.6804 0 0

a2

28.3183 27.1163 28.3183 28.3183 27.11631

a3

a4

-2.28 -2.42 -2.28 -2.28 -2.42

8.1820 8.3530 8.3530 9.0662 9.0662

ApPENDIX

8

Lifting Surface Correction Factors I

I

I!

A complete set of tables of the lifting surface correction factors (which are functions of the number of blades, th~ expandeil blade area ratio and the skew angle, well as the non~dimensional ra~ius and the advance ratio) would occupy too much space. Only three tables are given here for use with the problems given in the book.

as

Table AS.l I

Number of blades i= 3

I

Expanded blade area ratio = 0.8500

II

Skew angle = 0 degrees x 0.2000

\

0.3000

AI:

0.1000

0.2000

0.3000

0.4000

0.5000

0.6000

0.7000

ka : kc : kt :

4.0881 1.9633 2.4393

2.4545 1.2688 2.1302

1.1668 1.0742 1.8698

0.2252 1.3794 1.6583

-0.3704 2.1846 1.4956

-0.6200 3.4896 1.3818

-0.5237 5.2945 1.3167

k a'.

2.7960 1.4609 0.9068

2.3315 1.2802 0.5941

1.9668 1.2666 0.3594

1.7018 1.4202 0.2027

1.5366 1.7409 0.1240

1.4712 2.2288 0.1233

1.5055 2.8838 0.2006

kt :

2.1372 1.2192 0.5065

2.1309 1.3359 0.3627

2.0856 1.4265 0.2541

2.0013 1.4913 0.1808

1.8780 1.5300 0.1428

1.7157 1.5428 0.1400

1.5144 1.5297 0.1724

ka : k c'. kt :

1.6594 1.2382 0.2625

1.9082 1.4358 0.2136

2.0099 1.5539 0.1767

1.9645 1.5926 0.1520

1.7721 1.5519 0.1394

1.4326 1.4317 0.1388

0.9461 1.2321 0.1504

k' c'

kt

I

004000

:

k a'. k c'.

f.

t

I iI I:

I,

V

l

l

0.5000

485

BaBic Ship Propulsion

486

;;;.

§

Table A8.1 (Contd.) ka : kc : kt :

1.6879 1.2147 0.1295

1.8825 1.5719 0.1261

2.0145 1.7117 0.1237

2.0837 1.6341 0.1223

2.0902 1.3391 0.1219

2.0341 0.8266 0.1225

1.9152 0.0967 0.1241

ka

kt :

1.7113 1.6042 0.0733

2.0131 1.7956 0.0830

2.1133 1.9230 0.0901

2.0119 1.9865 0.0947

1.7089 1.9861 0.0967

1.2042 1.9218 0.0961

0.4980 1.7936 0.0930

0.8000

ka : k' c' kt :

1.8759 2.0898 0.0455

2.2440 2.1562 0.0575

2.3479 2.1864 0.0658

2.1878 2.1803 0.0704

1.7635 2.1379 0.0712

1.0751 2.0593 0.0684

0.1225 1.9444 0.0618

0.9000

ka

3.5358 2.9693 0.0061

2.9384 2.8880 0.0262

2.8928 2.7861 0.0415

3.3990 2.6637 0.0519

4.4571 2.5207 0.0574

6.0669 2.3571 0.0581

8.2285 2.1730 0.0539

0.6000

0.7000

:

k' c'

:

k' c' kt

:

~

~\

n ":"" }i

~~ "

OM ':'~;

..

~'.~

%

Table A8.2

\{ ::~

~i:i

=4 Expanded blade area ratio = 0.5000 Skew angle = 10 degrees Number of blades

x 0.2000

:.~

t

AI:

0.1000

0.2000

0.3000

0.4000

0.5000

0.6000

0.7000

ka :

2.5026 1.9198 0.7149

2.3229 1.8014 0.5356

2.1816 1.7290 0.3903

2.0786 1.7028 0.2790

2.0139 1.7226 0.2016

1.9875 1.7886 0.1582

1.9993 1.9006 0.1488

2.2003 1.2921 0.4565

2.1983 1.2883 0.3733

2.2008 1.2975 0.3033

2.2076 1.3197 0.2464

2.2189 1.3550 0.2028

2.2347 1.4033 0.1723

2.2548 1.4647 0.1549

1.9155 0.9709 0.2539

2.0267 1.0347 0.2409

2.1209 1.0910 0.2260

2.1981 1.1399 0.2093

2.2582 1.1815 0.1907

2.3013 1.2156 0.1704

2.3275 1.2423 0.1482

i

1.6481 0.9562 0.1074

1.8081 1.0405 0.1385

1.9421 1.1095 0.1586

2.0499 1.1633 0.1676

2.1317 1.2019 0.1656

2.1875 1.2253 0.1525

2.2172 1.2335 0.1284

I

1.5556 1.0260 0.0497

1.6800 1.1075 0.0850

1.7882 1.1727 0.1101

1.8801 1.2217 0.1252

1.9558 1.2545 0.1301

2.0152 1.2710 0.1249

2.0584 1.2713 0.1096

k' c' kt :

~

~ "

0.3000' k a : k· c' kt :

0.4000

k a'. k' c'

0.5000

0.6000

kt

:

ka

:

kc

:

kt

:

k a'. k c·. k t. :

.~

&

~

>;

~

~

i ~

~

I'

~

ii '1

}l

~.,

q

J

Review Questions 1. Describe the development of ocean transportation from prehistoric .times to the present day. 2. On a sketch of a screw propeller, indicate the following: boss, blade, . root"tip, face, back, leading edge and tr~iling edge. ,

I

3. The momentum theory of propellers is said to be based on correct fundamental principles while the blad~ element theory is said to rest on observed facts. Explain. !

4. 'What are the conditions that must be fulfilled in carrying out model tests with propellers? . 5. Explain how the hull and, the propeller of a ship interact with each other. \

6. ':Vhat is cavitation? How is cavitation affected by the temperature and the air content of water? 7. 'Vhat are the forces that are normally taken into account in calculating the stresses in a propeller blade? What are the factors that are not taken into account? 8. 'Vhat are the objectives of carrying out (a) resistance experiments, (b) open water experiments and (c) self-propulsion experiments with ship models and model propellers? 9. "'That is the main difference between the design of propellers for mer chant ships and the design of propellers for tugs and trawlers? 489

490


10. What are the tests and trials that are carried out on a ship before it is delivered to its owners? 11. What are the steps that can be taken to minimise propeller excited vibration in a ship? 12. What are the reasons for adopting unconventional propulsion devices in some ships? 13. Enumerate the different types of ships commonly used today. 14. Explain the concept of the pitch of a screw propeller. What is variable pitch? 15. Why is the propeller efficiency derived from the axial momentum the ory called an "ideal efficiency"? In which condition will this efficiency be 100 percent? 16. Why is it not possible to make the Reynolds numbers of a ship propeller and its model equal and simultaneously make the two Froude ~umbers equal too? Why are the Froude numbers made equal rather ~han the Reynolds numbers? 17. The wake of a ship has three component causes. Which of these com ponents would be absent if: (a) the ship was moving in an inviscid fluid, (b) the ship had an infinitesimal breadth, and (c) the ship was \moving deeply submerged? Can you think of a situation in which the wake fraction would be zero at all speeds? 18. How does ca\itation affect the performance of a marine propeller? 19. How are the stresses in a propeller blade affected by (a) the mass of the blade, (b) the propeller rpm, (c) the rake of the blade and (d) its skew, the thrust and torque being fixed? 20. How are the differences between the conditions of the model exper iment and the operating conditions of a ship taken into account in determining the effective power of the ship? 21. What are the two approaches to prop"eller design? Discuss their com parative advantages.

Review Questions

491

22. Why are the speed trials of a ship carried out? 23. How does the number of blades in a propeller affect the occurrence of unsteady forces at certain frequencies? 24. What are the advantages and disadvantages of using feathering paddle wheels instead of wheels with fixed paddles? 25. Discuss the development of ship propulsion machinery during the last two hundred years. 26. Explain the terms rake and skew with refElrence to a propeller blade. What is skew induced rake? What is warp? 27.. In tl;i.e axial momentum theory, the pressure in the slipstream far astern of the propeller is equal to the pressure .far ahead. Will this be true if the rotation of the slipstream is taken into account? 28. In an open water test with a model propeller, the thrust and torque coefficients are taken to be functions only of the advance coefficient, while the F'roude number, the Reynolds number'and the Euler number are neglected. How is this justified? 29. What is the difference between the nominal wake and the effective wake? 30. Describe the different types of cavitation and the conditions under which they occur. 31. What are the desirable properties in a material used for making pro pellers? 32. Why is it important to obtain turbulent flow in model experiments with ship models and model propellers? What steps are taken to ensure turbulent flow? 33. What are the considerations in selecting (a) the number of propellers in a ship, (b) the number of blades in a propeller, (c) the propeller diameter, (d) the rake of the propeller blades, and (d) the propeller blade skew? .\

492


34. If you were charged with carrying out the speed trials of a ship, how would you go about it? 35. Explain how an anti-singing edge eliminates the singing of a propeller. 36. In what types of ships would you consider the use of a controllable pitch propeller, and why? 37. What are the contributions made to ship propulsion by its pioneers: Colonel Stevens, Josef Ressel, Ericsson and Petit-Smith? 38. How would you
·. Review Questions

493

49. What are the different types of propulsion devices used in ships? 50. A propeller blade section is of a segmental shape with a chord c and a thickness t. What is its camber ratio? If the blade section were lenticular, what would be its camber? 51. Explain the concept of circulation and describe how it gives rise to lift in an aerofoil section. 52. What is a methodical propeller series? What are the parameters that are varied in a typical methodical series? 53. Why is the relative rotative efficiency not ~qual to 1 in general? 54. ,How is the cavitation of a propeller blade ~ection affected by the shape .of its mean line, the camber ratio, the thickness ratio and the angle of attack? 55. How do the blade sections of a super~avitating propeller differ from those of a conventional propeller? 56. Explain the concept of "ship self-propulsion point on the model". Why is a ship model not generally fully self-propelled in a self-propulsion experiment? 57. In propeller design why is the optimum propeller diameter determined from methodical series data sometimes reduced by a small amount? 58. Vlhat are the major factors that affect the propulsive performance of a ship in service? 59. \\'hat is the effect of surface roughness on the performance of a pro peller? 60. How do the open water characteristics of a surface piercing propeller differ from those of a conventional propeller? 61. Distinguish between nominal slip, apparent slip and effective slip. How is effective slip related to the effective pitch of a propeller? 62. Describe the vortex system of a wing of finite span. How is this vortex system related tel the vortex system of'a propeller blade?

494


63. Why are B p-8 diagrams more convenient to use in propeller design than Kr-KQ diagrams?

64. What are the components of the propulsive efficiency of a ship? How would these components be affected if the propeller were placed in front of the ship rather than behind it? 65. What are the dimensional parameters whose values must be carefully controlled in manufacturing a propeller? 66. Why are contra-rotating propellers more efficient than single propellers for the same thrust? 67. Why is it desirable to carry out direct wake measurements instead of merely determining the wake fraction through open water and self propulsion experiments? 68. What are the advantages and disadvantages of designing a tug proI peller for (a) the bollard pull condition and (b) the free running con dition? 69. What are the benefits of analysing the service performance of a ship? i

70. Describe how you would determine theoretically the time taken to stop a ship moving full speed ahea.d if its propeller is used for stopping the ship.

71. In,a vane wheel propeller, the vanes act as turbine blades at the inner radii and as propeller blades at the outer radii. Explain. 72. What are the major non-dimensional parameters used to describe a screw propeller? 73. Discuss the condition in which a propeller has the highest efficiency for a given speed of advance and revolution rate. 74. Why can B p-8 diagrams not be used for designing tug propellers? What type of diagrams are specially meant for designing tug pro pellers?

Review Questions

495

75. \Vhat do you understand by the effective power of a ship? Describe the stages by which the power produced by the main engine of the ship is transformed into the effective power. 76. Discuss the relation between the maximum rated output of the propul sion plant and the power for which the propeller is designed. 77. Outline the principal steps in designing a propeller using the lifting line theory. 78. Explain the concepts of "service margin" and "engine margin" on the . power of a ship p~opulsion plant. On what factors do these margins depend?

i

79. E:\..-P,lain the action of a Voith-Schneider j>ropeller, and show with the help of a sketch how it may be used to e'top a ship moving ahead. 80. \Vhat are the geometrical parameters ¥pon which the mass and polar moment of inertia of a screw propeller.depend? If in two geometrically similar propellers one has a diameter ,twice that of the' other, what will be the ratios of their masses and polar mo~ents of inertia? 81. Describe the recent developments in propeller theory. 82: Discuss the components of the overall efficiency of a waterjet propulsion system.

·I. " ~

,1

tl

L

Miscellaneous Problems 1. A ship with a resistance of 500 kN at a speed of 15 knots has a pro

peller whose thrust is 585 kN and torque 425 kN m. The shaft losses are 3 percent, the propeller rpm being 120. The propeller is estimated to produce a thrust of 585 kN in open water when running at 120 rpm at a speed of 12 knots, the corresponding torque being 450 kN m. De termine the effective power, the thrust power, the delivered power, the brake power, the wake fraction, the thrust deduction fraction, the hull efficiency, the propeller open water efficiency and the relative rotlLtive efficiency. What is the propulsive effici\mcy? 2. A ship is to haye a speed of 20 knots with its propeller of 5.0 m diameter running at 180 rpm, the depth of immersion of the propeller axis' being 4.9 m. At what speed of advance and at what pressure must a 0.125 m diameter model of the propeller be tested in a cavitation tunnel if the model propeller is to be run at 900 rpm? Wake· fraction = 0.250.. 3. A four-bladed propeller of 4.0 m diameter and 0.6 constant pitch ratio develops a thrust of 300 kN and a torque of 150 kN m at 180 rpm. The propeller has a blade thickness fraction of 0.045, the thickness varying linearly along the radius tc 12 rom at the blade tip. The root section is at O.2R and has an area given by 0./5et and a section modulus equal to 0.12et 2 , where e is the chord and t the thickness. The chord at O.2R has a length of 900 mm. Each blade of the propeller has a mass of 400 kg, its centroid being at a radius of 0.950 m and 0.100 m aft of the centroid of the root section. The thrust and torque per unit length increase linearly from a value of OAk at 0.2R to loOk at 0.5R, remain constant up to 0.9R, and Jecrease linearly to zero at 1.0R, k being a constant. Determine the tensile stress in the root section. 496

II I I

\

i

II

Miscellaneous Problems

497

4. A resistance experiment is carried out on a model of length 4.0 m and wetted surface 4.50 m 2 without appendages, and the following results obtained: Model speed, ms- 1 : 0.514 0.772 1.029 1.286 1.543 1.800 Model resistance, N: 2.594 5.486 9.663 15.397 26.029 42.0{)9 Determine for the corresponding speeds the resistance of a geometri , cally similar. ship of length 100 m using the ITTC friction line with a correlation allowance of 0.0004 and adding;10 percent to allow for appendages and air resistance. Neglect form factor. 5. A single-screw ship is required to have a Spel1d of 20 knots at which its effective power is 9500 kW. The wake fraction is 0.180, the thrust de duction fraction 0.145 and the relative rotative efficiency 1.030, based on thrUst identity. The maximum propelleJ;' diameter that can be fitted in the ship is 6.0 m, while the minimum eKpanded blade area required is estimated to be 0.700. The propeller js to have four blades and to be directly connected to the engine placed aft in the ship. Using the data of Table 9.2, determine the brake power and rpm of the engine, . and the corresponding pitch ratio of the propeller.

6. A ship has an engine of maximum continuous rating 9500 kW at 132 rpm directly connected to a propeller of 5.5 m diameter and 0.8 pitch ratio. Determine the maximum rpm at which the engine can be run during dock trials if (a) the torque is not to exceed 80 percent of the ma.ximum rated torque, (b) the maximum allowable longitudinal force on the ship is 800 kN, and (c) the propeller slipstream velocity is to be limited to 8.0 m s-1. The thrust and torque coefficients of the propeller are KT = 0.3415, 10KQ = 0.4021 at J = O. The thrust deduction fraction in the' static condition is 0.050 and the relative ro tative efficiency (thrust identity) is 1.030. The shafting efficiency is 0.970. The slipstream velocity may be calculated from the axial mo mentum theory. . 7. A ship of displacement 8750 tonnes has a propeller of 6.5 m diameter and 0.8 pitch ratio whose open water chara:cteristics are given by:

b


498

0.3415 - 0.2589J - 0.1656J2 10KQ

=

0.4021 - 0.2329J - 0.2101J 2

The wake fraction of the ship is 0.250, independent of speed, the thrust

deduction fraction is given by t = 0.100 + 0.007VK, where VK is the

ship speed in knots, and the relative rotative efficiency is 1.000. The

6 effective power of the ship is given by 0.1665 kW. The propeller

is directly connected to a diesel engine of 9000 kW brake power at

108 rpm, the shafting efficiency being 0.980. The ship has a design

speed of 18.00 knots. If the ship starts from rest, determine the time

taken by it to reach a speed of 17.99 knots and the distance travelled,

given that neither the maximum engine torque nor the maximum rpm

may be exceeded. The added mass is 5 percent of the ship's displace

ment. Carry out the calculations at 15 second intervals.

Vl

8. The expanded blade widths of a four-bladed propeller of 5.0 m diameter

are 'as follows:

R 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Blade width, mm: 989 1483 1760 1879 1978 1928 1701 1315 0

r/

Determine the expanded blade area ratio.

9. A

propeller of diameter 4.0 m operating 4.5 m below the surface of

.water has a speed of advance of 5.0 m per sec. The axial velocity of

'water in the slipstream is 2.5 m per sec far behind the propeller. Using

the axial momentum theory, determine the propeller thrust and the pressures and velocities in the slipstream relative to the propeller far ahead, just ahead, just behind and far behind the propeller.

10. A ship propeller of diameter 5.0 m running at 120 rpm and a speed

of advance of 6.0 m per sec has a thrust of 500 kN and a torque of

375 kN m. If a model propeller of 0.25 m diameter is to be tested at the

same Reynolds number and the same advance coefficient, determine its

speed of advance, rpm, thrust and torque. If the depth of immersion

of the ship propeller is 6.0 m, what should be the depth of immersion

of the model propeller? What will be the ratio of the Froude number

J


499

of the ship propeller to that of the model propeller and the ratio of the corresponding cavitation numbers? 11. A ship has a propeller of 5.0 m diameter. When the propeller runs at 120 rpm, it has a delivered power of 6000 kW and the ship has a speed of 16 knots at which its effective power is 4000 kW. The wake fraction is 0.280, the thrust deduction fraction 0.190 and the relative rotative efficiency 1.050, based on thrust identity. Calculate the hull efficiency, the propeller efficiency in open water, the propeller thrust and torque, the thrust power, and the advance, thrust and torque coefficients. 12. A ship has a propeller of 5.0 m diameter whose axis is 4.5 m below the load waterline. The ship has a speed of 20 knots at a propeller rpm of 180. The effective power of the ship at 20 knots is 7500kW, and the prop~ller efficiency in open water is 0.600. The wake 'fraction is 0.200, the thrust deduction fraction 0.150 and ~he relative rotative efficiency 1.020. A model of the propeller made to a scale 1:10 is to be tested in a cavitation tunnel at an rpm of 1200. , ' What should be the speed of water and the pressure in the cavitatioh tunnel? Determine the thrust atld torque of the model propeller. ' 13. A four-bladed propeller of 5.0 m diameter has the following distribution of pitch ratio and blade widths: rjR : 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0~9 1.0 PjD: 0.7000 0.7217 0.7425 0.7625 0.7817 0.8000 0.8175 0.8342 0.8500

cjD : 0.225 0.249 0.270 0.288 0.301 0.307 0.299 0.260 0

I

!

/,

I

l

The section modulus of the blade sections is given by 0.108ct 2 , where t is the thickness distributed linearly from root to tip, the tip thickness being 15 mm and the blade thickness fraction being 0.050. The pro peller has a speed of advance of 7.5 m per sec at 150 rpm, the delivered power being 6000kW and the thrust power 4000kW. The thrust and torque distributions are given by: dKT dx

~_

500

Basic Ship Propulsion Determine the stress due to thrust and torque at the different radii, taking the root section to be at 0.2R.

14. An open water experiment is carried out with a model propeller of 0.175 m diameter and 1.000 pitch ratio. The propeller has four blades, and at the section at 0.75R the chord-diameter ratio is 0.260 and the thickness-chord ratio 0.050. At the advance coefficient corresponding to the normal operating condition of the propeller, the following values are obtained:

=

Speed of advance

6.3 ms- 1

Thrust

Model propeller rpm

= 2700

Torque -

237.4 N 7.81O'Nm 't

Determine the advance coefficient, thrl,lst coefficient, torque coefficient and open water efficiency of the model propeller: If the ship propeller of diameter 4.9 m has the same thrust, torque and advance coefficients as the model propeller, estimate the surface roughness ofthe ~hip pro peller. . 15. A single screw ship has an engine of 15000 kW at 150 rpm directly connected to the propeller. The wake fraction, thrust deduction frac tion and relative rotative efficiency based on thrust identity are 0.180, 0.145 and 1.030 respectively, and the shafting efficiency is 0.980. The propeller axis is 5.5 m below the waterline. The effective power onhe ship at different speeds is as follows: Ship speed, knots Effective power, kW:

18.0 6570

19.0 7939

20.0 9500

21.0

11269

I 1 • \

!

Design the propeller using the data of Table 9.1. 16. A single-screw ship of length 90 m, wetted surface 1800 m 2 and above water transverse projected area 150 m 2 is taken out on trials and the following readings obtained during successive runs at constant power setting:

\ ~

I

I \

,.


501

Run No. Direction Time at start, hr-min Time on mile, min-sec Propeller rpm Shaft torque, kN m Relative wind speed, knots

1

N 15-15 4-2.8 143.3 198.2 3.3

2 S

15-41 4-4.6 144.3 201.8 32.8

3 N

4

S

16-06 16-33

4-0.8 4-7.1 143.2 144.8 199.2 202.4 3.4 32.9

The \\ind during the trials has no significant component transverse to .the ship. The slope of the effective power-speed curve for the ship speed under consideration is 512.4 kW per knot. The total resistance coefficient of the ship derived from model t~ts without any correlation allowance is as follows: . i

0.150 3.0659

0.200 3.8799

0.250 4.5900

0.300 5.1960

If the overall propulsive coefficient of the ship is taken as 0.710, deter mine the correlation allowance C A for the ship in the trial condition.

I

t

f

I

I

l

17. A single-screw tug has a propeller of 3.08 m diameter and 0.8 pitch ratio whose open water characteristics are given by:

KT = 0.3415 0.2589 J - 0.1656 J2

10KQ = 0.4021 - 0.2329 J - 0.2101 J2

The wake fraction is 0.250, the thrust deduction fraction in the free running condition is 0.210 and the relative rotative efficiency is 1.020, based on thrust identity. The effective power of the tug is given by PE = 0.01791 VI~·7kW, where VK is the speed in knots. The propeller is connected through a double input single output gearbox to two iden tical diesel engines each of which can be uncoupled from the propeller. The shafting efficiency is 0.950. If the propeller is to run at a constant 180 rpm over the complete speed range qf the tug, determine the min imum brake power of each engine. Determine also the speed of the


502

tug at which one of the engines can be uncoupled from the propeller without the maximum rated torque being exceeded. What is the free running speed of the tug and the brake power of the engine in this condition? 18. A propeller of diameter 5.0 m has a pitch ratio of 0.8. The expanded blade section of the propeller at 0.6R has a chord length of 1.750m with the leading edge being 0.600 m forward of the reference line. The propeller has a rake of 10 degrees aft. Calculate the coordinates of the leading and trailing edges at 0.6R in the projected outline. 19. A propeller of diameter 3.0 m has a delivered power of 263 kW at zero speed. What is its thrust on the basis of the axial momentum theory? 20. It is estimated that a model propeller of 200 mm diameter has open water characteristics as follows: .

KT - 0.206 - 0.165 J - 0.106 J2 lOKQ

=

0.239 - 0.139 J - 0.125 J2

Determine the maximum rpm and maximum speed of adva?,ce of the model propeller if the delivered power available from the drive motor is 1.0 kW and the open water test is to be carried out at the maximum rpm to cover the range of slip from zero to 100 percent. What is the J)1.aximum thrust to be measured? 21. A ship has a main engine of 10000 kW rated brake power at 120 rpm directly connected to a propeller of 5.0 m diameter. The ship has a wake fraction of 0.240, a thrust deduction fraction of 0.164 and a relative rotative efficiency of 1.050, based on thrust identity. The shaft losses are 2.0 percent. The ship attains a speed of 15.55 knots with the rated brake power and rpm. Its effective power at this speed is 6000 kW. Find the open water efficiency and the thrust and torque of the. propeller. 22. A ship is to have a propeller of 4.5 m diameter and 0.75 pitch ratio with its axis 4.0 m below the load waterline. The ship speed is estimated to

• Miscellaneous Problems

503

be 16 knots at which the effective power is 4000 kW and the propeller rpm 120. The wake·fraction is 0.280 and the thrust deduction fraction 0.190. Determine the blade area ratio of the propeller using the Burrill criterion for'merchant ship propellers. 23. A four-bladed propeller of 6.0 m diameter has the following distribution of pitch ratio and blade width:

Pt) =

0.655 + 0.240 x - 0.045 x 2

The blade thickness fraction of the propeller is 0.050, the blade thick ness varies linearly with radius, and at the pp the thickness is 20 mm. The blade sections have an area given by 0.7et and a section modulus given by 0.100ct2 , where 'C andt are the~hord length and thickness of the section. The propeller is made of Manganese Bronze of density 8300 kg per m 3 • The propeller has a speed of advance of 8.5 m per sec at 120 rpm, the delivered power being 7500 kW and the thrust power 5250 kW. The thrust and torque distributions are given by:

The root section is at 0.2R. Determine the stresses at the different radii, neglecting the bending moment due to centrifugal force.

I I

24. A ship has a propeller of 5.0 m diameter and 0.8 pitch ratio. At the service speed of 20 knots, the propeller rpm is 180 and the brake power 14700kW, while the effective power is 9500kW. The shafting efficiency is 0.970. A self-propulsion test carried out with a 1/25-scale model gives a wake fraction of 0.200, a thrust deduction fraction of 0.120 and a relative rotative efficiency of 1.050 based on thrust identity. Calculate the thrust, torque and rpm of the model propeller and the speed of the ship model for conditions corresponding to the ship self propulsion point. If the tow force applied to the model is 1.0 N, what is the resistance of the model at this speed? If in the open water test the model propeller is run at 1800 rpm, what should be its speed of advance, and what will be the corresponding thrust and torque?

L__


504

25. A single-screw tug is required to have a bollard pull of 20 tonnes and a free running speed of 15 knots at which its effective power is 625 kW. The wake fraction at 15 knots is 0.250, the thrust deduction fraction being 0.190. In the static condition, the thrust deduction fraction is 0.050. The relative rotative efficiency may be taken as 1.000 and the shafting efficiency as 0.950. The propeller diameter is 3.5 m. Determine the minimum brake power required and the corresponding propeller rpm and pitch ratio. Use the data of Table 4.3.

1:,C1

ij

"I

.~

~fj

:,

i';(~

i I~

\~

>,~ .;

26. A twin-screw ship with propellers of 4.80 m diameter and 0.9 pitch

ratio is taken out on speed trials and the following data recorded:

Run Direction Time at Time on Propeller Propeller Shaft Relative Wind No. start measured ·rpm thrust power wind direction mile speed oft' bow kW knots d~g hr-min sec kN 1 2 3 4 5 6 7 8 9' 10

SW

11

'SW

NE S\V NE

SW NE

SW NE

SW NE

12 . NE 13 SW

9-15 10-06 11-10 11-51 12-34 13-15 13-48 14-24 14-57 15-27 15-55 16-24 16-53

416.2 326.9 402.4 266.6 289.1 289.7 219.2 238.8 215.7 198.0 186.5 192.5 192.9

95.9 96.4 95.7 124.7 125.0 124.5 153.4 153.9 153.2 191.7 192.6 191.5 192.0

225.4 226.6 224.9 381.6 382.5 381.0 576.8 578.7 576.1 948.5 953.0 947.5 950.1

1785 1805 1780 3935 3940 3925 7310 7330 7275 14965 15035 14960 14990

13.58 5.15 18.66 3.87 25.15 1.09 29.40 3.22 28.35 8.09 28.66 12.37 23.77

1p.6

38.0 10.0 47.3 4.8 95.4 0.0 19.0 4.0 20.6 6.5 14.9 6.6

Determine the speed of the ship through the water for each of the four groups of runs and the speed of the current as a function of the time of day. Use Eqn. (10.3) for calculating the wind resistance. If the open water characteristics of the propellers are as given in Table 4.3, the shafting efficiency is 0.960 and the effective power of the ship is ~ given in the following, analyse the trials data to obtain the propulsion

., :"; :\:,

l·_'~ ;

t~


505

factors as a function of ship speed by both thrust identity and torque identity. Speed, knots Effective power, kW:

10 2044

12 3641

14 5458

16 8277

18 12881

20 20051

27. A propeller has a constant pitch ratio of 1.0 and a diameter of 3.0 m. Determine the pitch angles from 0.2R to the blade tip at intervals of O.IR.

28. A pI:'opeller of diameter 5.0 m has a speed of advance of 6.0 m per sec at 120 rpm. Its axial inflow factors from root to tip are as follows: !

r/R: a

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0.127 0.194 0.237 0.265 0.2S2 0.294 0.303 0.309 0.313

Using the impulse theory and neglecting drag, determine the thrust and torque of the propeller. Is the propeller working close to its optimum efficiency? 29. A propeller of 5.0 m diameter and 1.0 pitch ratio has the following open water characteristics: \

KT = 0.4250 - 0.2517 J - 0.1441 J2 10KQ = 0.5994 - 0.2733 J - 0.2254 J2 The propeller absorbs a delivered power of 7500 kW at 150 rpm. De termine the speed of advance, the thrust and the open water efficiency of the propeller. The relative rotative efficiency is 1.000.

,

I

L

30. A ship has a total resistance of 500 kN at its design speed of 15 knots. The engine, directly connected to the propeller, has a brake power of 5500 k\V at 150 rpm. The shafting efficiency is 0.970 and the propulsion factors are: wake fractiot. 0.200, thrust deduction fraction 0.150 and relative rotative efficiency 1.040. CalCulate the effective power, the

506

Basic Ship Propulsion thrust power, the delivered power, the propeller thrust and torque, the open water efficiency, the hull efficiency and the propulsive efficiency.

31. A ship has a draught of 6.0 m. Its propeller of 4.0 m diameter and 0.9 pitch ratio has its axis 3.5 m above the base line of the ship. The propeller develops a thrust of 300 kN at 180 rpm and a speed of advance of 6.0 m per sec. Determine the blade area ratio using the Burrill criterion for merchant ship propellers. 32. A four-bladed propeller of 6.0 m diameter and 0.8 pitch ratio has a delivered power of 7500 kW at 120 rpm. The propeller is made of Manganese Bronze of density 8300 kg per m 3 . The root section of the propeller at 0.2Rhas a chord of 1350 mm, the maximum chord length of the propeller being 1900 rom. The propeller blades have a 15 deg rake aft. The blade thickness fraction is 0.050, the blade thickness at the tip being 20 mm. Determine the tensile and compressive stresses using Taylor's method. I

33. A 4.0 m long ship model with a wetted surface of 3.60 m 2 has aresis tance of 35 N at a speed corresponding to 20 knots for the ship, whose length is 100 m. A self-propulsion test is carried out on the model and the following results obtained at this speed: Propeller rpm:

500

550

600

650

700

27.15

30.84

35.83

42.22

50.81

Torque, Nm :

0.61

0.93

1.42

2.08

2.91

Tow Force, N :

6.33

5.47

4.65

3.70

2.63

Thrust, N

'The open water characteristics of the model propeller, which has a diameter of 0.200 m, are as follows: 0.700

0.800

0.900

1.000

0.2551

0.2242

0.1913

0.1534

0.508

0.468

0.420

0.368

-~


507

Calculate the effective power, the delivered power and the propeller rpm of the ship at 20 knots in the trial condition using the ITTC ship performance prediction method, given the following values':

l+k

-

1.065

6CF -

-0.0002 6KQ 6KT 6w c = 0.005

0.4

X

10- 3 CAA = 0.7

0.0003

b"CFC

X

10-3

= 0.3}( 10-3

\\That are the predicted values of wake fraction, thrust deduction frac tion and relative rotative efficiency of the ship in the trial condition? 34. A single screw tug has an engine of 1500 kW at 480 rpm connected to a propeller of diameter 3.5 m through 3 : 1 reduction gearing, the shafting efficiency being 0.950. The depth of immersion of the pro peller shaft centre line is 3.3 m. The wake fraction is 0.250, the thrust deduction fraction varies linearly with speed being 0.050 at zero speed and 0.190 at 15 knots, and the relative rotative efficiency is 1.000. The effective power of the tug at different speeds is as follows: Speed, knots Effective power, kW:

8 69.2

10 151.2

12 14 286.2 ·490.9

16 783.4

18 1183.1

Show that the required expanded blade area ratio is less than 0.500. Using the data of Table 4.3, determine the limits within which the pitch ratio must lie so that the bollard pull is not less than 20 tonnes and the free running speed not less than 15 knots. 35. The fine weather data extracted from a record of the service perfor mance of a ship are given in the following table. Analyse these data to determine the effect of days out of dry dock on the power of the ship at a displacement of 15000 tonnes and a speed of 17.0 knots. Estimate the brake power at 15000 tonnes displacement and 17.0 knots speed at zero and 400 days out of dry dock.

No. 1.

~.

1Ia-

gs

Displacement tonnes

Speed knots

Days out of dry dock

Brake power kW

14100

17.5

25

6860

508

Basic Ship Propulsion 2.

3. 4.

5. 6. 7. 8. 9. 10.

15300 15800 14700 14900 15300 14500 15900 14700 15200

16.9 16.6 17.2 17.1 16.8 17.5 16.3 17.3 16.3

80 100 150 175 210 250 290 320 360

6650 6470 6935 6910 6710 7360 6360 1250 6820

36. A propeller of diameter 5.0 m has a constant geometric face pitch ratio of 1.0. The blade section at 0.7R, which may be regarded as rep resenting the whole propeller, is of aerofoil shape such that the line joining the leading and trailing edges ("nose-tail line") makes an angle of 1.5 degrees with the face chord. The no-lift angle with respect to the nose tail line is 2.0 degrees.· Determine the effective pitch ratio: of the propeller. 37. A four-bladed propeller of diameter 5.0m and constant face pitch ratio 0.8 is advancing into undisturbed water at a speed of 6.0 m per sec at 120 rpm. The blades are composed of NACA-66 (modified) sections for which:

9£ =

0.1097(1-0.83tlc)(a+2.35)

CD =

0.0100

a in degrees

+ 0.0125 cl

The blade widths and thicknesses at the various radii are as follows: 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 r I R: 0.2 clD: 0.208 0.235 0.256 0.269 0.273 0.268 0.246 0.198 0 tic : 0.176 0.138 0.110 0.089 0.072 0.058 0.046 0.036

Calculate the thrust, torque and efficiency of the propeller, neglecting induced velocities.


509

38. A propeller is required to produce a thrust of 350 kN at 120 rpm at a speed of advance of 12.0 knots. Using the Bp-8 diagram, Fig. 4.6, determine the optimum diameter and pitch ratio of the propeller, and the corresponding delivered power in open water. 39. A ship has a design speed of 15 knots, its resistance at this speed being 500 kN. The ship has a propeller of 5.0 m diameter directly connected to an engine of brake power 5500 kW at 120 rpm. The propeller is designed to operate at J = 0.620, KT = 0.230, 10KQ = 0.350. If the shaft losses are 3 percent, determine the propulsive efficiency and its components based on thrust identity. Determine also the effective power, the thrust power and the delivered power. 40. A four-bladed propeller of diameter 6.0 m and pitch ratio 0.8 has a . brake power of 7500 kW and a thrust power of 5250 kW at an rpm of 120 and a speed of advance of 8.5 m per sec. The propeller has a bla9.e thickness fraction of 0.050, the tip thickness being 20 mm. The blades are raked aft at 15 deg and have a normal blade outline. The expanded blade area ratio of the propeller is 0.550. The propeller is made of Manganese Bronze of density' 8300 kg per m 3 • The root section is at 0.2R, and has an area of 23000mm2 and a section modulus of 8.05 x 106 mm 3 • Determine the stress in the root section using Burrill's method. 41. In a self propulsion test at the ship self propulsion point on a 1/25 scale model at a speed of 2.0 m per sec, the propeller thrust and torque are 23.04 Nand 0.728 N m respectively with the model propeller of 200 mm diameter running at 800 rpm. The resistance of the model is 25.00 N and the resistance of the ship at the corresponding speed is 315 kN. The model propeller has a thrust of 23.04 N and a torque of 0.760 N m when running at 13.333 revolutions per sec at a speed of advance of 1.60 m per sec in open water. Calculate the tow force applied to the model, and determine the wake fraction, the thrust deduction fraction and the relative rotative efficiency. Calculate the effective power and the delivered power of the ship. 42. A twin-screw tug has two engines each of brake power 500 kW at 1200 rpm connected to the propellers of 2.00 m diameter through 4 : 1 reduction gearing. The effective power of the tug is as follows:

L


510


10 153.7

12 300.0

14 528.2

16 862.3

The wake fraction is 0.100, the thrust deduction fraction varies lin

early from O.O~O at zero speed to 0.120 at 16 knots, and the relative

rotative efficiency is 0.980 based on thrust identity. The shafting effi

ciency is 0.950. The depth of immersion of the propeller shaft centre

line is 1.8 m. Determine the expanded blade area ratio required for

the propellers, and calculate the pitch ratio that will give the maxi

mum bollard pull with a minimum free running speed of 12.50 knots.

Calculate the engine rpm and brake power in the bollard pull and free

running conditions. Use the data of Table 4.3.

43. A method of determining the effect of weather on the service perfor mance of a ship is to plot PE /n 3 and V/n as functions of a ''weather intensity number" I'V, where PE is the brake power, n the propeller rpm and V the ship speed. W may be taken to be the product of the wind speed squared and a "wind direction factor" , equal to 1.0 for bow winds (0-45 degrees off the bow), 0.5 for beam winds (45 to 135 off the bow) and 0.2 for stern winds (135-180 degrees off the bow). Analyse by this method the following data for a ship (which have been corrected to the fully loaded, clean hull condition) and determine the brake power and ship speed at the rated propeller rpm of 119 in fine weather (W = 0).

I 1

I

I

~

Miscellaneous Problems 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.

14.61 12.04 13.15 7.25 9.96 6.67 6.55 5.12 11.42 7.83 9.27 8.00 7.75 11.55 12.07 12.10 11.05

511 115.1 114.6 113.3 96.9 92.2 79.3 76.3 68.5 97.9 76.3 100.4 91.9 98.6 94.6 108.3 107.7 109.8

6410 6330 6020 4910 3390 2650 2280 1690 4070 1830 4770 3920 4840 3650 5600 5530 5670

15 38 28 42 44 40 41 45 23 23 45 38 38 28 33 32 26

160 60 70 5 170 20 110 140 145 35 85 35 35 '175 .65 60 30

44. A propeller of constant pitch ratio 0.8038 has blades that ·are raked aft at an angle of 15 deg. Determine the skew angles at r / R = 0.2,0.3,0.4, ... ,0.9, 1.Q for the blade reference line to lie in the blade reference plane normal to the propeller axis. 45. A four-bladed propeller of diameter 4.0 m has a speed of advance of 7.5 m per sec at 150 rpm. The ideal thrust of the propeller is 400 kN. Determine the ideal thrust loading coefficient. If for this value, the ideal efficiency is 0.800 and the blade width distribution is as given in the following table, determine the lift coefficients at the different radii.

r/R c/D

0.2 0.3 0.225 0.249

0.4 0.270

0.5·· 0.288

0.6 0.301

0.7 0.307

0.8 0.299

0.9 0.260

1.0 0

The Goldstein factors may be determined from Appendix 6. 46. A propeller of 4.0 m diameter and 0.8 pitch ratio has a delivered power of 2000 kW at 180 rpm. Using the 11,-(T diagram, Fig. 4.7, determine the speed of advance and the thrust of the propeller. If the torque is


512

constant, calculate the rpm, thrust and delivered power at 0, 5, 10 and

15 knots speed of advance.

47. A ship has a speed of 15 knots when the engine produces 5000 kW brake power at 180 rpm. The effective power of the ship is 3600 kW. The propeller is directly connected to the engine. The wake frac: tion is 0.250, the thrust deduction fraction 0.150, the relative rotative efficiency 1.030 and the shafting efficiency 0.980. Determine the de livered power, the thrust power, the overall propulsive efficiency, the propulsive efficiency (quasi-propulsive coefficient), the hull efficiency, the open water efficiency, the speed of advance, the resistance of the ship, and the propeller thrust and torque. 48. The axial velocities measured at various angular and radial positions

in the plane of the propeller of a single-screw ship model moving at a

speed of 3.0 m per sec are as follows: Angular position, deg

0

Radius, mm 20 30 40 50 60 70 80 90 100

30

60

90

150'

180

1.943 1.981 2.034 2.103 2.186 2.286 2.400 2.529 2.674

1.775 1.820 1.881 1.960 2.057 2.172 2.305 2.454 2.622

120

<

i~

I t ~,

Axial velocity, m per sec 1.250 1.313 1.402 1.515 1.653 1.817 2.006 2.221 2.460

1.593 1.644 1.714 1.806 1.917 2.049 2.201 2.373 2.566

1.972 2.009 2.061 2.128 2.209 2.305 2.416 2.542 2.683

2.096 2.128 2.174 2.233 2.304 2.389 2.487 2.597 2.721

2.147 2.178 2.221 2.276 2.344 2.423 2.516 2.620 2.737

Determine the average circumferential wake fraction at each radius and

the average wake fraction over the propeller disc. The model propeller

has a diameter of 200 mm and a boss diameter ratio of 0.200.

49. Design a propeller for a single screw ship using the circulation theory

(lifting line theory with lifting surface corrections). The design speed

of the ship is 16.0 knots at which the effective power is 3460 kW. The

J


513

wake fraction is 0.200 and the thrust deduction fraction 0.150. Prelim inary design calculations indicate a propeller with four blades, 5.200 m diameter, 0.500 expanded blade area ratio, 0.045 blade thickness frac tion and a skew angle at the blade tip of 10 pegrees. The propeller rpm is 126 and the immersion of the shaft axis is 3.5 m. The effective wake fractions at the different radii are as follows: x = r / R: 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 w 0.305 0.295 0.279 0.259 0.235 0.206 0.174 0.136 0.095

The expanded blade outline is to con:form to the B-series, and the Lerbs criterion for wake-adapted propelle~s is to be used. Determine the delivered power and the propeller efficiency in the behind condition. ,

.

.

I

50. A ship is fitted with an engine of 5500 kW brake power at 120 rpm directly connected to a propeller of 5.0m diameter. The ship has a speed of 15 knots and a resistance of 500 kN. The propeller operates at J = 0.625, KT = 0.220, lOKQ = 0.330.' The relative rotatiye efficiency is 1.060. Determine the propulsive effiCiency and its compcinents based on torque identity, and the effective power, the delivered power and the shafting efficiency.

\

,

L

Answers to Problems Chapter 1

Chapter 2 1.

Arc length of projected outline = 404.9 mm

Arc length of developed outline = 480.0 mm

2.

Face pitch = 3.996 m.

3.

Width of expanded outline

Pitch ratio = 0.7992

= 720.0 mm

Distance of reference line from leading edge = 288.0 mm Pitch ratio = 0.864 4.

Expanded blade area ratio

= 0.6499

Developed blade area ratio = 0.6455 Projected blade area ratio = 0.5667 5.

Thickness chord ratio = 0.1488 Camber ratio

6.

\I

= 0.0250

~

Mean pitch ratio = 0.9716 Pitch at 0.7 R

= 5.892 m 514

\ 1

J

515

Answers to Problems 7.

Speed of advance for zero slip

= 11 ms- 1

Speed of advance for 100 percent slip = 0

8.

Pitch ratio

= 0.9000

Blade area ratio (expanded) Blade thickness fraction

= 0.4400

= 0.0510

Boss diameter ratio = 0.1670

9.

Diameter = 1.624m

= 0.2838

Projected blade area ratio = 0.2006

E:-..rpanded blade area ratio

Mean pitch ratio = 2.1227

10.

11ass

= 4961.1 kg

Polar moment of inertia

= 4235.55 kg m 2

Chapter 3 1. Location

Pressure) kN m- 2

= 130.4 kN

2.

Thrust

3.

Diameter = 2.056 m

4.

Delivered power

5.

Thrust

Efficiency

1

5.000 6.244 6.244 7.487

45.249 38.083 53.999 45.249

Far ahead Just ahead Just astern Far astern

~lBIIIIIilII

Velocity, ms

= 0.750

= 262.4kW

= 222.58kN

Torque = 132.44kNm

Efficiency = 0.8559

_

516

Basic Ship PropulBion

6.

Thrust = 503.10kN

Torque

= 320.28kNm

7.

Thrust = 1263.94kN Torque

= 665.08kNm

8.

-x -

0'0

fie

PID

0.2 0.4 0.6 0.8 1.0

1.5725 1.3309 1.0210 0.8341 0.7521

0.0095 0.0080 0.0062 0.0050 0.0045

0.8514 0.8454 0.8440 0.8426 0.8457

9.

x

---

dTddx

dQi/dx

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

96.623 231.254 404.473 589.995 783.274 919.680 986.480 897.800

100.335 240.852 421.260 614.481 815.785 957.849 1027.422 935.061

Efficiency

= 0.9074

Delivered power = 4908 kW

10.

x , 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

c CL D 0.08452 0.09558 0.09428 0.08726 0.07818 0.06702 0.05430 0.03833 0

,

\ \

I ~

J

Answers to Problems

517

Chapter 4 1.

Speed of advance = 1.342 m s-1

Rpm = 536.66

Thrust = 60.976N

Torque = 2.287Nm

Ratio of cavitation numbers = 0.0761 Ratio of Reynolds numbers 2.

= 85.756

Diameter = 0.2000 m

Speed of advance = 1.789 ms- l

Rpm = 670.71

Thrust= 15.61ON

3.

J

KT'

10KQ

1]0

0 0.1111 0.2222 0.3333 0.4444 0.5556 0.6667 0.7778 0.8889 1.0000

0.4250 0.3948 0.3611 0.3240 0.2834 0.2393 0.1917 0.1406 0.0860 0.0280

0.6100 0.5711 0.5280 0.4806 0.4289 0.3730 0.3129 0.2485 0.1799 0.1070

0 0.1222 0.2419 0.3577 0.4674 0.5673 0.6500 0.7004 0.6763 0.4165

Torque = 0.5854Nm

Effective pitch ratio = 1.0514

4.

Diameter = 4.854m

5.

Rpm

= 122.3

Pitch ratio = 1.150 Efficiency

= 0.630

Thrust = 704.6 kN Pitch ratio

= 1.241

Delivered power = 2852.8 kW

6.

l

1~4.,

knots: n, rpm T, kN PD,kW

~

o 88.54 61.74 202.23

2.5 98.37 58.36 224.71

.5.0 7.5 112.23 129.39 54.38 49.46 256.35 . 295.52

10.0 149.34 43.20 341.13

13.5 180.00 31.39 411.14


518

7.

Diameter = 4.608 m

Pitch ratio = 0.972

Deliyered power = 6040 kW

8.

Rpm = 1-14.3

9. -

Pitch ratio = 0.800

Pitch ratio = 0.920

VA, knots:

0

3.0

6.0

9.0

12.0

n, rpm

150

158

173

190

210

PD,kW

1000

1056

1155

1264

1400

176

171

161

152

141

T, kN 10.

Diameter

=

4.052 m

Pitch ratio = 0.5

Thrust = 622.08 kN

Thrust =221.5 kN

(Lower limit of pitch ratio)

For diameter = 3.50 m:

Pitch ratio = 0.751

Thrust = 197.4 kN

Chapter 5 1.

Effective power = 4526.72 kW \

Torque

= 467.92kNm

Thrust = 687.50 kN

Thrust power = 4243.80kW

, Open water efficiency = 0.7007


Overall propulsive efficiency = 0.7545

2.

Effective power = 4115.20 kW

Thrust pO\l;er = 3703.68 k\V


Hull efficiency

= 1.1111


= 1.0796

Open ,vater efficiency = 0.5895

Propulsive efficiency = 0.7071

...,

].

Answers to Problems

5.

x

t,mm: 6.

0.2 161.4

521 0.3 140.0

0.4 117.6

0.5 96.3

0.6 0.7 75.8 56.0

= 40.208 N mm

2

7.

Blade thickness fraction = 0.06495

8.

Blade thickness at 0.25R

= 189.806 mm

Blade thickness at 0.60R

= 83.176 mm

10.

0.9 18.0

Blade thickness fraction = 0.04638 Stress in root section

9.

0.8 36.7

Blade thickness fraction

= 0.0496

Blade thickness fraction

= 0.0488

Bending moment about x-axis = -543.083 kN m Bending moment about y-axis Torsion about radial axis

= 433.330 kN m

= -289.613kNm

Tensile force on the root section

=

913.562 kN

Chapter 8 1.

Speed, knots Effective power, kW

10.0 551

12.0 1010

14.0 1771

16.0 3048

18.0 5198

2.


12.0 1787

14.0 3106

16.0 5291

18.0 8926

20.0 14953

3.

(a) Diameter

= 0.1111 m Reynolds number = 3.932 X 105 at zero slip Reynolds number = 3.574 X 105 at 100 percent slip = 0.1246 m Reynolds number = 3.958 X 105 at zero slip Reynolds number = 3.598 X 105 at 100 percent slip

(b) Diameter

~(

1.0


522 (c) Maximum rpm = 1508.5

Reynolds number = 3.604 x 105 at zero slip Reynolds number = 3.277 x 105 at 100 percent slip. 4.

J KTS 10KQS 'TJos

J KTS 10KQs 'TJos

0,0000 0.3399 0.3893 0.0000

0.1000 0.3123 0.3701 0.1343

0.2000 0.2812 0.3454 0.2591

0.3000 0.2470 0.3150 0.3744

0.5000 0.1700 0.2377 0.5691

0.6000 0.1261 0.1906 0.6318

0.7000 0.0797 0.1378 0.6444

0.8000 .0.9000

0.0302 -0.0228

0.0794 0.0151

0.4843

5.

. Thrust Identity

Torque Identity

0.2538 0.1549 1.1325 0.6179 1.0601 0.7418

0.1998 0.1549 1.0561 0.6505 1.0799 0.7418

Wake Fraction Thrust deduction fraction Hull efficiency Open water efficiency Relative rotative efficiency \ropulsive efficiency Ship speed = 19.44 knots Propeller rpm = 180 6.

Effective power = 7444kW Delivered power = 10035kW

= 0.1304 Hull efficiency = 0.9808

Thrust deduction fraction::: 0.1471

Wake fraction


0.4000

0.2103

0.2793

0.4793

Open water efficiency :=

= 0.7392

1.0339

Propulsive efficiency = 0.7496 These results in general are not applicable to the ship since they have been obtained at the model self propulsion point and not at the ship self propulsion point on the model.

Answers to Problems

5.

X

t, mm: 6.

521

0.2 0.3 161.4 140.0

0.4 117.6

Blade thickness fraction Stress in root section

0.5 96.3

0.6 0.7 75.8 56.0

0.8 36.7

0.9 18.0

1.0

= 0.04638

= 40.208 N mm

2

7.

Blade thickness fraction = 0.06495

8.

Blade thickness at 0.25R = 189.806 mm Blade thickness at 0.60R = 83.176 mm Blade thickness fraction

9. 10.

= 0.0496·

Blade thickness fraction = 0.0488 Bending moment about x-axis = -543.083 kN m Bending moment about y-axis =

433~330 kN m

Torsion about radial axis = -289.613 kN m Tensile force on the root section = 913.562 kN

Chapter 8 1.


10.0 551

12.0 1010

14.0 1771

16.0 3048

18.0 5198

2.


12.0 1787

14.0 3106

16.0 5291

18.0 8926

20.0 14953

3.

(a) Diameter

= 0.1111 m

Reynolds number

= 3.932 x

Reynolds number

=

(b) Diameter Reynolds

3.574 x 105 at 100 percent slip

= 0.1246 m number = 3.958 x

Reynolds number

105 at zero slip

105 at zero slip

= 3.598 x 105 at 100 percent slip

~---------------


522 (c) Maximum rpm

= 1508.5

Reynolds number == 3.604 x 105 at zero slip

Reynolds number 4.

J

= 3.277 x 105 at 100 percent slip.

KTS 10KQ8 1]OS

0.0000 0.3399 0.3893 0.0000

0.1000 0.3123 0.3701 0.1343

0.2000 0.2812 0.3454 0.2591

0.3000 0.2470 0.3150 0.3744

J J(TS 101(Q8 1]08

0.5000 0.1700 0.2377 0.5691

0.6000 0.1261 0.1906 0.6318

0.7000 0.0797 0.1378 0.6444

0.8000 .0.9000

0.0302 -0.0228

0.0794 0.0151

0.4843

5.

Thrust Identity

Torque Identity

0.2538 0.1549 1.1325 0.6179 1.0601 0.7418

0.1998 0.1549 1.0561 0.6505 1.0799 0.7418

Wake Fraction Thrust deduction fraction Hull efficiency Open water efficiency Relative rotative efficiency Propulsive efficiency , Ship speed == 19.44 knots Propeller rpm = 180 6.

0.4000 0.2103 0.2793 0.4793

Effective power = 7444kW Delivered power == 10035 kW

== 0.1304 Hull efficiency == 0.9808

Wake fraction


== 0.1471

Open water efficiency = 0.7392

Relative rotative efficiency == 1.0339 Propulsive efficiency == 0,7496 These results in general are not applicable to the ship since they have been obtained at the model self propulsion point and not at the ship self propulsion point on the model.

Answers to Problems

7.

523

Thrust deduction fraction = 0.1531

\Yake fraction = 0.2127 Hull efficiency

= 1.0757

Open water efficiency

= 0.6170

Relative rotative efficiency = 1.0545 Propulsive efficiency = 0.6999 Delivered power = 7215 kW Propeller rpm 8.

Kominal wake fraction = 0.1959

9.

r. mm:

lL'(r)

15.0

22.5

30.0

37.5

45.0

= 126.5

52.5

60.0

67.5

75.0

: 0.069 0.075 0.083 0.092 0.098 0.110 0.122 0.134 0.150

Kominal wake fraction = 0.1118 10.

Speed of water = 1.7893 ms- 1

Model propeller rpm = 948.6

1!odel propeller torque = 0.3032 N m Pressure = 3.7631 kN m- 2 I

Chapter 9 1.

Total shaft power

= 22407kW

Propeller rpm = 243.98

Pitch ratio = 1.5255 2.

3.

4.

Diameter = 3.470 m


Blade area ratio = 0.8506

Speed = 35.212 knots

Diameter = 2.733 m




Speed, knots 15.0 20.0 25.0 30.0 35.0 40.0 35.157 'TUrbine rpm 1158 1589 2035 2495 2987 3414 3000 Shaft power, kW: 1111 3068 6774 12974 22685 35993 23037 Ship speed at 3000 rpm of turbines Shaft power = 23037 kW

L

= 35.157 knots


524 5.


Brake power

= 4712.4kW

Rpm = 118.75 '

6.

Diameter

= 3.416 m

Pitch ratio

= 0.5 (minimum in Table4.3)

Bollard pull = 169.43 kN

Free running speed

= 10.036 knots

Brake power = 300.3 kW

Rpm = 600

7.

Pitch ratio

= 1.1257

Bollard pull = 109.83 kN

Brake power = 616.9 kW 8.

Diameter = 2.829 m

Engine rpm Pitch ratio

= 411.28

= 0.5 (minimum of Table 4.3)

Maximum tow rope pull = 140.6 kN

Free running speed

= 11.640 knots


'Irawling with full cll.tch:

Speed = 5.628 knots


Engine rpm = 240 9.

Speed, knots

0.0

2.0

9.643

6.0

8.0

:row rope pull, kN: 132.74 128.18

120.85 110.71

96.41

Engine rpm

912.8

954.6

1006.2 1067.2 1136.9 1200.0

Brake power, kW :

380.3

397.8

10.0

12.0

Speed, knots

Tow rope pull, kN:

Engine rpm

72.95

4.0

419.2

444.6

473.6

80.97~

500.0

14.0 12.859

23.51 -33.51

0.00

1200

1200

1200

1200

Brake power, kW : 482.4

378.0

264.4

330.3

Speed at which engines develop maximum brake power = 9.643 knots Free running speed

= 12.859 knots

':, ,

, ~,

,


525

x

PID

elD

tiD

fie

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

1.2109 1.2396 1.2332 1.2280 1.2244 1.2201 1.2174 1.2189 1.1915

0.4629 0.5123 0.5567 0.5941 0.6212 0.6324 0.6154 0.5364

0.0406 0.0359 0.0312 0.0265 0.0218 0.0171 0.0124 0.0077 0.0030

0.0106 0.0154 0.0141 0.0137 0.0127 0.0131 0.0122 0.0113

Brake power = 12863 kW

Chapter 10 1.


Tens/ion in mooring rope = 99.15 kN

Reaction = 51.32 kN 2.

Run No.

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

L

_

Direction

N S N S N S N S N S N S

Time at Mid-run

Speed through water

Speed of current to North

hours

knots

knots

8.895 9.696 10.459 11.184 11.908 12.577 13.240 13.854 14.438 15.016 15.568 16.097

9.317 9.525 10.593 10.725 11.657 11.634 12.765 12.933 13.967 14.153 15.526 15.855

1.870 1.615 1.431 0.993 0.625 0.227 -0.242 -0.535 -0.785 -0.967 -1.068 -0.920


526

I .. ~

'

~

Runs No.

Average Speed knots'

Average Power kW

Average Rpm

Average Thrust kN

9.421 10.659 11.646 12.849 14.060 15.690

1491 2006 2555 3340 4212 5125

88.1 97.9 106.4 116.7 125.9 136.8

244 293 343 406 462 525

1 and 2 3 and 4 5 and 6 7 and 8 9 and 10 11 and 12

3.

Thrust Identity

V, knots

w

t

'fJH

'fJO

'fJR

'fJD

9.421 10.659 11.646 12.849 14.060 15.690

0.2641 0.2600 0.2586 0.2533 0.2513 0.2508

0.2048 0.1998 0.1979 0.1959 0.1900 0.1913

1.0792 1.0814 1.0818 1.0769 1.0819 1.0795

0.5707 0.5803 0.5833 0.5889 0.5955 0.6065

1.0544 1.0527 1.0538 1.Q502 1.0282 1.0529

0.6494 0.6607 0.6650 0.6661 0.6625 0.6893

'fJH

'fJO

'fJR

'fJD

1.0718 '1.0693. 1.0714 . 1.0670 1.0375 1.0718

0.6494 0.6607 0.6650 0.6661 0.6625 0.6893

Torque Identity

V, knots

w

t

9.421 10.659 11.646 12.849 14.060 15.690

0.2218 0.2212 0.2194 0.2175 0.2314 0.2169

0.2048 0.1998 0.1979 0.1959 0.1900 0.1913

1.0206 '0.5937 1.0275 0.6013 1.0274 0.6041 1.0277 0.6074 1.0539 0.6059 1.0327 0.6228

j;' ,~

~:-,

j,

~

Answers to Problems

4.

5.

527

Run No.

Speed knots

Shaft Power kW

1 to 3 4 to 7 8 to 10 11 to 13

13.995 15.874 19.254 20.807

2327.5 3622.5 7370.0 9827.5

Run No.

Time of day hours

1 2 3 4 5 6 7 8 9 10 11 12 13

8.099 8.804 9.499 10.300 10.896 11.698 12.292 12.997 13.675 14.394 15.107 15.693 16.308

Speed knots 13;995 15.874 19.254 20.807

6.

Ship speed

10 3 CA

0.260 0.349 0.354 ·0.365

= 18.324 knots

Propeller rpm

73.075 83.625 104.375 114.338

Speed of current knots

2.71p 2.707 , 2.698 2.307 1.944 ~.581

1.129 0.611 0.093 -0.399 -0.630 -0.915 -1.255 WTrial - WModel

-0.0218 -0.0195 -0.0223 -0.0215 Brake power = 16957 kW

Rpm = 113.969 The ship fulfils the contract.

L

_

528 7.

Basic Ship Propulsion End of Year No.

Speed

Propeller Rpm

knots 17.008 16.819 16.638 16.464 16.296 16.135 15.979 15.829 15.579 15.303 15.036

0 1 2 3 4

5 6 7 8

9 10

126.0 126.0 126.0 126.0 126.0 126.0 126.0 126.0 124.9 123.8 122.5

Brake Power

Maximum Brake Power Available

kW

kW

8815 8960 9099 9232 9362 9484 9602 9715 9573 9384 9189

10500 10395 10290 10185 10080 9975 9870 9765 9573 9384 9189

The propeller must be changed at the end of the eighth year. i

Pitch ratio of new propeller = 0.6584 Ship speed = 15.135 knots 8.

Increase in power I %

Wave direction Head seas , Bow quartering seas , Stern quartering seas Following seas

5.4 3.9 -2,4 -7.6

i 9.

Days in sen'ice

Corrected power I kW

10 120 310

9597 98G8 10139

f ~

Ii

Ship Drydocked 390

535

9792 10092

I I I

.-1

529

Answers to Problems

10274

665 Ship Drydocked

750 890 1050

9925 10231 10458 Ship Drydocked

10063 10358 10579

1110 1235 1385 Days in service

PoWer just after drydocking kW

o

9573 9723 9873 10023 10173 10323

365 730 1095 1460 1825

10.

Days after drydocking i

Increase in Effective Power

Propeller rpm

% 0 10 20 30 40 50

144.0 144.0 143.5 142.3 141.4 140.5

Increase in power kW

o

o

60 , 120 180 240 300 360 365

154 286 399 491 564 616 619

Brake power

Ship speed

kW

knots

10123 10386 10461 10379 10309 10246

17.677 17.122 16.647 16.164 15.737 15.356

:

I

L

_

530


Chapter 11 The 6-bladed propeller should be selected.

1.

The 5th , 6 th and 7th harmonics in the wake velocities are small.

2.

Increase in effective power over that in calm water

Ship· speed

%

knots

o

18.019 17.158 16.425 15.795 15.246

15 30 45 60 3.

kW 150.00 147.50 145.38 143.59 142.05

9989 9834 9692 9572 9469 Brake power

= 9000 kW

After one year: Ship speed = 17.5 knots Rpm = 119.4

Brake power

= 8272 kW

Time to stop

= 207 s

Distance travelled

\

5.

Brake power

Ship speed = 19.0 knots Rpm = 126.0

New

4.

Propeller rpm

= 912.9 m

'rrial condition:

Speed = 18.730 knots Rpm = 118 Brake power= 13486 kW

Average service condition:


Rpm = 118


I

Service margin = 5.56 percent

Engine margin = 5.37 percent

Ma.ximum permissible increase in effective power over that in the

trial condition = 49.83 percent

Corresponding ship speed

= 16.668 knots

r

Answers to Problems

531

Chapter 12 1.

= 0.230 m

Dist&nce of feathering point forward of centre

Diameter of fixed paddle wheel of same entry angle

2.

Speed of 'lUg

Tow rope Pull

Pitch Ratio

kN

knots

o 2 4

6

8 10 12 14 15.573

= 65.824 m

161.962 150.024 136.176 119.939 101.023 : 79.177 54.110 25.430

0.7883 0.8205 0.8590 0.9038 0.9545 1.0109 1.0724 1.1383 1.1928

o

Free running speed == 15.573 knots

3.

4.

Bollard pull = 239.678 leN

Free running speed::: 12.617 knots


Brake power

Engine· rpm == 600

Engine rpm = 600

Nozzle thrust::: 128.118 kN

Nozzle thrust

Thrust::: 27.106 kN

Torque

Horizontal force ==4.937kN

Vertical force

= 623.95kW = 8.402 kN.

= 2.9943 kN m =0

Effective power::: 275.4kW 5.

Bollard pull (with both propellers)

= 63.312 kN

Delivered power per ·propeller ::: 99.16 kW 6.

....

H

Speed::: 40.0 knots

Pump efficiency::: 0.918


532 Miscellaneous Problems:

1.

Effective power' = 3858.00 kW

Thrust power = 3611.09kW

Delivered power = 5340.71 kW Brake power = 5505.88 kW Wake fraction = 0.2000 Hull efficiency


= 1.0684

= 0.1453

Open water efficiency- = 0.6386

Relative rotative efficiency = 1.0588 Overall propulsive efficiency = 0.7007 2.

Speed

= 0.9645 ms- 1

3.

Stress

= 31.921 Nmin-: 2

4.

Ship speed, knots: Resistance, leN

Pressure = 3.836 kN m- 2

'

5.0

7..5

10.0

12.5

26.11 . 57.74 106.98

15.0

179.03 332.95 595.78

5.

Brake power = 14287kW

6.

Maximum permissible engine rpm: (a) 96.2, (b) 97.3, (c) 93.6

7.

Time taken to reach a speed of 17.99 knots = 240 s

rpm = 127.2

Distance travelled = 1807 m

\

8.

.Expanded blade area ratio = 0.650

9.

Thrust

= 201.258 kN

Location

Pressure

Velocity

kNm- 2

ms- 1

Far ahead

45.249

5.000

Just ahead

38.042

6.250

Just behind

54.057

6.250

Far behind

45.249

7.500

'';t

17.5

Pitch ratio ='0.970


533

Speed of advance = 115.0505 ms- 1

Rpm = 46020

Thrust = 448.395 kN

Torque = 16.8148 kN m

Depth of immersion of model propeller = 0.30 m Ratio of Froude numbers = 0.01166 Ratio of cavitation numbers =559.4305 11.

12.

\

13.

1

Hull efficiency = 1.1250

Open' water efficiency = 0.5644

Thrust = 600.00 kN

Torque == 477.46 kN m

Thrust power == 3555.6 kW

J = 0.5926

K T = 0.2341

10KQ = 0.3913

Speed of water = 5.487ms- 1

Pressure == 64.521 kNmm- 2

Thrust = 3.719 N

Torque == 0.2653 N m

r/

R

: 0.2

0.3

0.4

0.5

I

0.6

0.7

0.8

0.9

Stress: 30.627 29.856 27.966 25.562 22.777 19.379 15.004 8.621 Nmm- 2 14.

110del propeller:

J = 0.8000

KT = 0.1250

10KQ = 0.2350

7]0

= 0.6773

Ship· propeller roughness = 46 11m 15.

Diameter = 5.798 m


Expanded blade area ratio = 0.6807 Speed = 20.162 knots 16.

Correlation allowance = 0.3 x 10- 3

17.

Rated brake power of each engine = 1000 kW Speed at which one engine may be uncoupled == 13.653 knots Free running condition: Speed = 16.00 knots Brake power ,= 759.7 kW.

L

1.0

534


18.

Cartesian coordinates in mm:

Leading edge:

1399.41, 540.00, -30.08

Trailing edge:

1141.63, -972.98, 713.78.

19.

Thrust = 100.077 kN

20.

Maximum rpm

,.

~:;.

= 1650.35

Maximum speed of advance

= 4.5016 m 8 1

Maximum thrust = 0.2494kN


21.

Torque

Thrust = 986.975 kN

= 779.859 kN m

22.

Blade area ratio

23.

r/R

: 0.2

= 0.9065

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

I

Stress: 51.496 34.141 26.04721.617 18.74417.19415.618 12.604 Nmm- 2 24.

Self propulsion test: Model prcpeller: Thrust = 65.519 N

Torque

= 1.889 N m

Rpm = 900 \

Speed of ship model

= 2.0576 m s-1

Model resistance

= 58.657 N

Open water test: Speed of advance Torque 25.

Pitch ratio

Run No.

1 to 3

Thrust = 262.080N

= 7.557Nm

= 0.830

Propeller rpm 26.

= 3.292ms- 1


= 147.9

Speed through water

knots

9.956

-

.

Answers to Problems Run No.

535

Speed through water knots 12.887 15.935 18.947

4 to 6 7 to 9 10 to 13

Run No.

Time at mid-run

Speed of current

hours

knots

9.3078 10.1454 11.2226 11.8870 12.6068 13.5415 13.8304 14.4332 14.9800 15.4775 15.9426 16.4267 16.9101

-1.2289 -1.0929 -0.8524

1 2 3 4 5 6 7 8 9 10 11 12 13

-0.6345 -0.1027 0.4292 0.7454 0.8857 1.0259 0.7186 0.4550 0.1608 -0.1641

Thrust Identity

V, knots

w

t

1]H

7]R

1]0

1]D

9.956 12.887 15.935 18.947

0.0589

0.1333 0.1363 0.1380 0.1409

0.9209 0.9161 0.9176 0.8973.

0.9823 0.9826 0.9847 0.9752

0.6436 0.6431 0.6434 0.6315

0.5822 0.5788 0.5814 0.5527

0.0572 0.0607 0.0426

==


536 Torque Identity

27

V, knots

w

t

9.956 12.887 15.935 18.947

0.0718 0.0700 0.0718 0.0629

0.1333 0.1363 0.1380 0.1409

r/R

.,::>

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

57.86 46.70 38.51 32,48 27.95 24,45 21.70 19,48 17.66

28. \ Thrust

7]H .

0.9338 0.9287 0.9286 0.9167

= 503.251 kN

7]R

7]0

7]D

0.9738 0.9742 0.9770 0.9652

0.6403 0.6398 0.6408 0.6246

0.5822 0.5788 0.5814 0.5527


The propeller is working close to its optimum efficiency (0.7500).

29.

Speed of advance

= 9.961 m s-l

Thrust

= 532.258 kN


30.

Thrust power

Delivered power = 5335 k\V

Propeller thrust = 588.325 kN

= 339.637kN m Hull efficiency = 1.0625


Torque

31.

= 3631kW

Effective power = 3858 kW


Propulsive efficiency

= 0.7231

.

;


537

Tensile stress = 55.908 N mm- 2 Compressive stress = 60.334Nmm- 2

33.

Effective power = 5360.5 kW

Delivered power == 7445.3 kW

rpm = 122.98

Wake fraction = 0.2054

Thrust deduction fraction = 0.1400 Relative rotative efficiency = 1.0846 34.

Pitch ratio limits: 0.7465 to 0.7993

35.

For 15000 tonnes displacement and 17.0 knots speed: Days out of dry dock

Brake Power, kW

o

6940 7041

400 36.

Effective pitch ratio = 1.167

37.

Thrust = 1379.764 kN


Efficiency = 0.9094 38.

Pitch ratio = 0.925

Diameter = 4.779 m Delivered power = 3453 kW

39.

Propulsive efficiency = 0.7232

Effective power = 3858 kW

Hull efficiency == 1.0559

Thrust power == 3654 kW

Relative rotative efficiency = 1.0563

Delivered power == 5335 kW

Open water efficiency = 0.6484 40.

Stress = 45.524Nmm- 2

41.

Tow force = 5.332 N


\Vake fraction = 0.2000

Effective power;:;:; 3150 kW

== 1.044

Thrust deduction fraction = 0.1463- Delivered power =4883.7 kW

538


42.

Required expanded blade area ratio = 0.4872 Pitch ratio = 0.6673 Brake power = 2 x 391.48 kW

Bollard pull = 136.31 kN Engine rpm = 939.6

Brake power-= 2 x 306.39kW

Free running condition:

Engine rpm = 1200 43.


Speed = 14.61 knots

44.

r/

45.

Ideal thrust loading coefficient = 1.1042

R: 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 (}, deg: -12.0 -18.0 -24.0 -30.0 -36.0 -42.0 -48.0 -54.0 -60.0

r/R: CL :

r/R: CL :

0.2 0.2729

0.3 0.2974

0.4 0.2787

0.5 0.2475

0.7 0.1866

0.8 0.1599

0.9 0.1341

1.0

Speed of advance = 9.1365 m 8- 1

46.

0.6

0.2166

= 17.761 knots

Thrust = 123.4 kN \

47.

Speed of advance, knots: Rpm Thrust, kN Delivered power, kW

o

5 108.47 201.33 1205.3

93.74 214.17 1041.3

Delivered power = 4900kW

10 131.65 176.83 1462.7

15 161.87 145.00 1798.6

Thrust power = 3176.5kN

Overall propulsive efficiency = 0.7200 Propulsive efficiency = 0.7347 Hull efficiency = 1.1333

Open water efficiency

Speed of advance = 5.787 ms- 1

Resistance

Thrust

= 548.898 kN

= 0.6294

= 466.563 kN


I ,

J


r

539

w(r)

rom

20 30 40 50 60 70 80 90 100

0.3742 0.3608 0.3419 0.3176 0.2881 0.2529 0.2125 0.1668 0.1154

Average wake fraction 49.

= 0.2449

Final blade geometry;

x

PID

fie

e/D

0.8093 0.8011 0.7993 0.8004 0.8071 0.8163 0.8265 0.8353 0.8430

0.0487 0.0329 0.0228 0.0196 0.0179 0.0170 0.0170 0.0191

0.2078 0.2353 .0.2562 0.2690 0.2734 0.2680 0.2462 0.1978 0

tiD

;

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0366 0.0324 0.0282 0.0240 0.0198 0.0156 0.0114 0.0072 0.0030

Delivered power = 4864kW

Propeller efficiency (behind) = 0.6741

50.

= 0.7261 Open water efficiency = 0.6631 Thrust deduction fraction = 0.1633 Effective power = 3858 kW Propulsive efficiency

Hull efficiency = 1.0330

= 0.1900 Shafting efficiency = 0.9660

Wake fraction

Delivered power = 5313 kW

,

-7;!

:.'I \'

, II

I

Bibliography and References Books

1. Abbott, L.H. and Doenhoff, A.E. von (1959), "Theory of Wing Sections", Dover, New York, U.S.A. 2. American Bureau of Shipping (19'99)', "Rules and Regulations for' the Classification of Ships", New York, U.S.A. 3. Breslin, J.P. and Andersen, P. (1994), "Hydrodynamics of Ship Propellers", Campridge University Press, Cambridge, U.K. ' I

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"I

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Abbreviations DTMB

David Taylor Model Basin, Carderock, U.S.A.

lESS

Transactions of the Institution of Engineers and Shipbuilders of Scotland, Glasgow.

IMarE

Transactions of the Institute ()f Marine Engineers,L'Ondoll.

542

Basic Sbip Propl!'lsion

INA

Institution of Naval Architects, London.

ISP

International Shipbuilding Progress, Delft, The Netherlands.

JSR

Journal of Ship Research (pUblished by the Society of Naval Architects and Marine Engineers, New York).

MT

Marine Technology (pUblished by the Society of Naval Archi tects and Marine Engineers, New York).

NACA

National Advisory Committee on Aeronautics, U.S.A.

NECIES

Transactions of the North East Coast Institution of Engineers and Shipbuilders, Newcastle upon Tyne, U.K. . ). .

RINA

Transactions of the Royal Institution of Naval Architects, London. .

SNAJ

Proceedings of the Society of Naval Architects of Japan, Tokyo.

SNAME

TransaCtions ofthe Society of Naval Architects and Marine En gineers, Jersey City. .

SSPA

Statens Skepps Provings Anstalt ([Swedish] State Ship T~ting Establishment), Goteborg, Sweden. '

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